And what is this reason for the similarities?
none? other than "woah that's cool!"
it's science fiction. The themes of the story have many metaphors and visions that apply to modern culture and human society in general, but there doesnt have to be a reason for everything.
Bioshock just takes place in a world where the same exact story is being continuously told in an infinite (wink) number of different universes. Only with different contexts.
I mean sure, it also works as fan service to take you there, but it also helps in showing what Elizabeth means by "another city, another man". The trip to Rapture is meant to give you another chance to go "ohhh" in case you didnt get it at first. There's nothing IN there that shows you this but the trip alone is intended to make you think of the game.
And the similarities are not just similarities for the sake of it, it is very clear that this is the absolute intention.
Not only by Elizabeth's line, but you can do a silly run down on each universe's equivalents:
Rapture = Columbia
both meant to be an "ark" of sorts for what their leader considers the elite of humanity, meant to live outside of plebe society for one rason or the other
Andrew Ryan = Comstock
both leaders driven mad by their ambitions above
Big Daddy = Songbird
both mutant creatures meant to take care of an asset that is integral to the success of the ark
Little Sisters = Elizabeth
the assets that need protection. Both successfully "rebel" in the end (remember: the little sisters are the one that ultimately kill Fontaine, same way Elizabeth is the one who kills booker/comstock here)
Frank Fontaine/Atlas = Daisy Fitzroy
people who raise up to challenge the established society but are revealed to corrupt themselves or have a much deeper and darker meaning behind their actions that relate them more to the person they oppress
Brigid Tenenbaum = Lutece twins
Scientists working for the leader who develop technologies that are crucial to achieve the society's goal. Both end up rebeling and aiding the player throughout the journey.
it is absolutely in your face throughout the game mayne