In short, probably not, and I am here to explain why.
Also, this thread is obviously going to be chock full of spoilers, so consider yourself warned.
Now then, I have been hearing about this theory more and more lately, as people have been drawing parallels between characters in the Bioshock games in accordance with the multiverse revelations. Now, let me reach for my analytical glasses and see what we can figure out.
First off, let us see the hypotheses.
-#1: Booker is an alternate universe Andrew Ryan
-#2: Elizabeth is an alternate universe Andrew Ryan
-#3: Elizabeth is an alternate universe Brigid Tenenbaum
-#4: Literally everyone is mirroring someone, including Rapture=Columbia
Also, before we jump into this, let us have a quick overview on how the multiverse works (some details interpolated in accordance of the Many Worlds Theory):
-Each time there is a variable (a decision point, in our case) with multiple outcomes, it creates a new branch with that a new parallel universe.
-Elizabeth's power can tear down the walls between universes, sort of merging them, or rather letting one flow into another.
-The Luteces can also do that, but they are also scattered across time as well, and are being able to view the entire tree. Elizabeth gets to this point by the end as well.
-Parallel universes ALWAYS have a branching point from which they originate from and time flows at the same rate in all of them. That means that there is no such thing as two parallel universes with different years.
Now, let's see those hypotheses one by one.
#1: Booker is an alternate universe Andrew Ryan
-The Argument: The comparison is a given from the very beginning, especially if you consider that he is also Comstock and whatnot. However, the most obvious evidence linking them together was from the ending, when we got a bit of a cameo appearance out of Rapture. Based on the state of affairs, it was obvious it was after the lock-down (as evidenced by the protest-signs everywhere), so the bathyspheres were supposed to be locked as well, yet Booker could use them. As we all (hopefully) know, the bathyspheres were DNA locked, and to Andrew Ryan's DNA, to be precise, so it would be immediately obvious that, if Booker could use it, he MUST be Ryan, right?
-The Rebuttal: Well, there are two huge problems with this. First off, you should not forget that Ryan was, in fact, Russian, unlike Booker. Now, you could argue that this is a thing that could be changed by the branching of the multiverse, but remember that there still must be a common point for the branching on the first place, and we know Ryan's background well enough to see there is simply no point where they could intersect.
Which also brings us to the second problem: the time-line. Remember, we know for a fact that Booker was born in 1874, and while we don't know when Ryan was born, we do know that he died in 1960. No matter how I look at it, Ryan doesn't look like a man in his late eighties to me in Bioshock. Also, before someone brings it up: no, the time-slip cannot be explained by multiverse, the universes are different because of the choices people made in them, they doesn't have different time-scales.
#2: Elizabeth is an alternate universe Andrew Ryan
-The Argument: If Booker is not a good enough candidate, then what about Elizabeth. We were shown that the same person can have different genders in different universes (look at the Luteces), and if Ryan was an alternate version of her, it could still explain how Booker could use the bathysphere: the same way Jack did, by being closely related genetically.
-The Rebuttal: This time around the age fits a tiny bit better, though still far from perfect, taken that she was born in 1892, so Ryan must have been born at the same time, which means he would have been almost 60 by the time Rapture was built, which is, again, quite a bit of a stretch. However, this time it is his nationality that really kicks into this hypothesis, as Anna is 100% surely born in America, and cannot be born anywhere else or otherwise there would be no branching. You simply cannot get around the fact that Anna and Ryan have such vastly different backgrounds, something the even the multiverse's possibilities cannot hand-wave away.
#3: Elizabeth is an alternate universe Brigid Tenenbaum
-The Argument: She is the most important female character in Rapture and her little sisters kinda-sorta dress like Elizabeth (if you squint really hard).
-The Rebuttal: This is just silly. Really, all the previous problems apply; neither her age, nor her nationality matches, and the "similarities" are strenuous at best.
#4: Literally everyone is mirroring someone, including Rapture=Columbia
-The Argument: Look at all the similarities! Megalomaniac leader of a secluded, technically advanced city, whose right hand man is a industrialist bigwig, taken down by his own close relative after/during a revolt by the masses led by a charismatic but crazy extremist.
-The Rebuttal: Well, duh. Of course there are similarities like that, otherwise this wouldn't be the same series. However, there are just as many differences as similarities, and you really cannot call it anything more than the developers simply working with familiar concepts. Call me when you discover something really outstanding.
So, here comes the final question:
"But if they are not related, then how could Booker operate the bathysphere?"
Well, here are a few possibilities:
-By some strange coincidences, one of Booker's ancestors (or Booker himself) emigrated into Russia, had a boy there who for some reason wasn't a DeWitt, who then grew up to be Ryan, thus they are still genetically related. I personally find this a wee bit too convoluted.
-The other way around: DeWitt's father could have been a Russian immigrant and thus this is how they are related. This is even less likely though, as Booker is implied to be part native American instead...
-Maybe the bathysphere was broken?
-Maybe it's not "our" Rapture. Maybe it is from an universe where Ryan was taken down before he could initiate the lock-down.
-Or, and this is my best bet, the dev team simply didn't really care about this detail. After all, it is a pretty minuscule thing, right? It's not some ill guy who had nothing better to do would analyze the shit out of it while bedridden, right?
...
Wait...
......
Oh... -.-'
------
Source: The Bioshock wikia, especially the time-lines, plus my own playthroughs.
Also, this thread is obviously going to be chock full of spoilers, so consider yourself warned.
Now then, I have been hearing about this theory more and more lately, as people have been drawing parallels between characters in the Bioshock games in accordance with the multiverse revelations. Now, let me reach for my analytical glasses and see what we can figure out.
First off, let us see the hypotheses.
-#1: Booker is an alternate universe Andrew Ryan
-#2: Elizabeth is an alternate universe Andrew Ryan
-#3: Elizabeth is an alternate universe Brigid Tenenbaum
-#4: Literally everyone is mirroring someone, including Rapture=Columbia
Also, before we jump into this, let us have a quick overview on how the multiverse works (some details interpolated in accordance of the Many Worlds Theory):
-Each time there is a variable (a decision point, in our case) with multiple outcomes, it creates a new branch with that a new parallel universe.
-Elizabeth's power can tear down the walls between universes, sort of merging them, or rather letting one flow into another.
-The Luteces can also do that, but they are also scattered across time as well, and are being able to view the entire tree. Elizabeth gets to this point by the end as well.
-Parallel universes ALWAYS have a branching point from which they originate from and time flows at the same rate in all of them. That means that there is no such thing as two parallel universes with different years.
Now, let's see those hypotheses one by one.
#1: Booker is an alternate universe Andrew Ryan
-The Argument: The comparison is a given from the very beginning, especially if you consider that he is also Comstock and whatnot. However, the most obvious evidence linking them together was from the ending, when we got a bit of a cameo appearance out of Rapture. Based on the state of affairs, it was obvious it was after the lock-down (as evidenced by the protest-signs everywhere), so the bathyspheres were supposed to be locked as well, yet Booker could use them. As we all (hopefully) know, the bathyspheres were DNA locked, and to Andrew Ryan's DNA, to be precise, so it would be immediately obvious that, if Booker could use it, he MUST be Ryan, right?
-The Rebuttal: Well, there are two huge problems with this. First off, you should not forget that Ryan was, in fact, Russian, unlike Booker. Now, you could argue that this is a thing that could be changed by the branching of the multiverse, but remember that there still must be a common point for the branching on the first place, and we know Ryan's background well enough to see there is simply no point where they could intersect.
Which also brings us to the second problem: the time-line. Remember, we know for a fact that Booker was born in 1874, and while we don't know when Ryan was born, we do know that he died in 1960. No matter how I look at it, Ryan doesn't look like a man in his late eighties to me in Bioshock. Also, before someone brings it up: no, the time-slip cannot be explained by multiverse, the universes are different because of the choices people made in them, they doesn't have different time-scales.
#2: Elizabeth is an alternate universe Andrew Ryan
-The Argument: If Booker is not a good enough candidate, then what about Elizabeth. We were shown that the same person can have different genders in different universes (look at the Luteces), and if Ryan was an alternate version of her, it could still explain how Booker could use the bathysphere: the same way Jack did, by being closely related genetically.
-The Rebuttal: This time around the age fits a tiny bit better, though still far from perfect, taken that she was born in 1892, so Ryan must have been born at the same time, which means he would have been almost 60 by the time Rapture was built, which is, again, quite a bit of a stretch. However, this time it is his nationality that really kicks into this hypothesis, as Anna is 100% surely born in America, and cannot be born anywhere else or otherwise there would be no branching. You simply cannot get around the fact that Anna and Ryan have such vastly different backgrounds, something the even the multiverse's possibilities cannot hand-wave away.
#3: Elizabeth is an alternate universe Brigid Tenenbaum
-The Argument: She is the most important female character in Rapture and her little sisters kinda-sorta dress like Elizabeth (if you squint really hard).
-The Rebuttal: This is just silly. Really, all the previous problems apply; neither her age, nor her nationality matches, and the "similarities" are strenuous at best.
#4: Literally everyone is mirroring someone, including Rapture=Columbia
-The Argument: Look at all the similarities! Megalomaniac leader of a secluded, technically advanced city, whose right hand man is a industrialist bigwig, taken down by his own close relative after/during a revolt by the masses led by a charismatic but crazy extremist.
-The Rebuttal: Well, duh. Of course there are similarities like that, otherwise this wouldn't be the same series. However, there are just as many differences as similarities, and you really cannot call it anything more than the developers simply working with familiar concepts. Call me when you discover something really outstanding.
So, here comes the final question:
"But if they are not related, then how could Booker operate the bathysphere?"
Well, here are a few possibilities:
-By some strange coincidences, one of Booker's ancestors (or Booker himself) emigrated into Russia, had a boy there who for some reason wasn't a DeWitt, who then grew up to be Ryan, thus they are still genetically related. I personally find this a wee bit too convoluted.
-The other way around: DeWitt's father could have been a Russian immigrant and thus this is how they are related. This is even less likely though, as Booker is implied to be part native American instead...
-Maybe the bathysphere was broken?
-Maybe it's not "our" Rapture. Maybe it is from an universe where Ryan was taken down before he could initiate the lock-down.
-Or, and this is my best bet, the dev team simply didn't really care about this detail. After all, it is a pretty minuscule thing, right? It's not some ill guy who had nothing better to do would analyze the shit out of it while bedridden, right?
...
Wait...
......
Oh... -.-'
------
Source: The Bioshock wikia, especially the time-lines, plus my own playthroughs.