I don't really understand the whole idea that the characters have to be the same person between universes, just equivalents. Booker isn't necessarily Jack, he's just his equivalent when taking the two stories together. The two lighthouses aren't the same lighthouse, they are just equivalents between the two stories. The same can be said of Rapture and Columbia, they are just two equivalent cities serving the same purpose, but they are not the same city.
BioShock Infinite commented on a lot of things: quantum mechanics, the nature of a depressed (and possibly suicidal) man, American nationalism and religiously-fueled politics, and time and our relationship to it. Outside of maybe the character analysis of Booker, I find that the only way to really get all of its ideas to come together into one coherent story is if we see it as a commentary on the nature of storytelling, most notably in relation to video games. Elizabeth's comments towards the end sort of establishes part of this: when dealing with a video game series, there will always be certain constants and variables among the games. In the BioShock universe, you will always have a man, a city, and a lighthouse; but there will also be differences among the many games. For instance, Rapture and Colombia are both unbelievable cities built around extreme ideologies, but one is built underwater and the other built in the sky. Rapture is based on staying away from the surface and keeping it that way, while Colombia only plans on remaining distant for a time before raining death down on the world below. If you apply this to pretty much everything in the games you can find these constants and variables, like comparing Booker to Jack, Comstock to Ryan, the Lutece "twins" to Tenenbaum and Suchong, Elizabeth to the Little Sisters (including Eleanor from BioShock 2), or Fitzroy to Fontaine. This idea also becomes more fleshed out if you start looking at franchises other than the BioShock series and applying its logic to those franchises. The commentary on storytelling in sequels goes especially well with an analysis of The Legend of Zelda.
BioShock Infinite commented on a lot of things: quantum mechanics, the nature of a depressed (and possibly suicidal) man, American nationalism and religiously-fueled politics, and time and our relationship to it. Outside of maybe the character analysis of Booker, I find that the only way to really get all of its ideas to come together into one coherent story is if we see it as a commentary on the nature of storytelling, most notably in relation to video games. Elizabeth's comments towards the end sort of establishes part of this: when dealing with a video game series, there will always be certain constants and variables among the games. In the BioShock universe, you will always have a man, a city, and a lighthouse; but there will also be differences among the many games. For instance, Rapture and Colombia are both unbelievable cities built around extreme ideologies, but one is built underwater and the other built in the sky. Rapture is based on staying away from the surface and keeping it that way, while Colombia only plans on remaining distant for a time before raining death down on the world below. If you apply this to pretty much everything in the games you can find these constants and variables, like comparing Booker to Jack, Comstock to Ryan, the Lutece "twins" to Tenenbaum and Suchong, Elizabeth to the Little Sisters (including Eleanor from BioShock 2), or Fitzroy to Fontaine. This idea also becomes more fleshed out if you start looking at franchises other than the BioShock series and applying its logic to those franchises. The commentary on storytelling in sequels goes especially well with an analysis of The Legend of Zelda.