I think it's mostly just a test to see if people were paying attention and to spark questions just like this. Also quite probably something they intended to figure more into the storyline later but never developed, but left the choice in. It's also possible it will in some way influance DLC. Basically the instinct in such cases is to go with "Bird" because of what it normally symbolizes, when every clue, NPC hint, etc... points you in the counter-intuitive direction.
I'd also point out another side of this that a lot of people don't seem to bother to think through. Is freedom for Elizabeth REALLY a good idea? If she was to ever get free of what constrains her and managed to still retain control of her powers she could represent a threat to the fundemental fabric of reality, a thousand times greater than someone like Comstock ever conceived of being. I've oftentimes wondered if the symbolism with the cage fits into that possibility, and if some of the DLC might deal with the concept of what she might turn into (like Booker into Comstock) if ever given free reign to pretty much do whatever the hell she wanted with that power.
The thing is that by making Elizabeth a cute girl and likable, we're sympathetic, we view her as a person can can easily take the whole "everyone deserves a chance to be free" arguement. On the other hand if you were dealing with some Lovecraftian horror with the abillity to travel between and overlap worlds saying "hey, I'd like to be free, I'm really a nice guy despite all these tentecles and a tendency for people to go insane just looking at me" we'd doubtlessly go "no.. just no... you stay locked up". Another more contemporary example of where I'm going with this would be say the twist in "The Ring" where it turns out that Samara, the girl your lead to sympathize with as an "avenging spirit" was basically some equivilent to The Antichrist and she was trapped in a well and left to die for good reason.
One big question in fantasy has always been between the whole X-men school of thought with "It's wrong to persecute or try and control something just for being powerful", vs. the whole "nobody should have power like that" school of thought where good guys always have to give it up or inevitably wind up becoming evil when they are free to do with it what they want over a prolonged period of time. Not to mention the whole Lovecraftian thing where from a certain moral perspective leaving them imprisoned could be considered wrong because it is their world, and human existance as we know it WAS a mistake (lol) which is the whole cosmic horror of the entire thing.
That's going beyond my point here, which is basically that I suspect your intended to stop and think this one through, and especially if your someone with a civilized western perspective (US, UK, etc...) have to start contemplating a point that goes contrary to what your being taught.