I haven't seen them say anywhere that it definitely isn't in the same universe, just that it has nothing to do with Bioshock in the sense of story and design, and different themes and a different atmosphere doesn't mean it can't hold the same name or be a spiritual sequel/prequel either.
Look at Shadow of the Colossus, they never confirmed it being a prequel until a few months after it came out, and you only find out it is till the very last scene of the game.
Though if they have confirmed it definitely isn't, a link to the source wouldn't go a miss.
Ken Levine neither confirmed nor denied if the two Bioshocks are related in story, I guess that's the big internet debate, huh?
I'm on the side that it's not. And I'd rather have it if they weren't related. I would like to see Bioshock Infinite being a spiritual successor of Bioshock just like Bioshock was the spiritual successor of System Shock.
Following this story, it seems as if Irrational is going for something completely unique for us to experience as players, something more than what is expected from the Bioshock brand. That's why I'm assuming that adding the Bioshock name is more marketing than anything else.
Some quotes to add to the ambiguity:
"For an audience, I totally expect them to be like, 'What? I don't quite understand it,'" Levine said. "'It's a sequel but is it a sequel?' It's a meta-question. I'm comfortable with that." He won't say whether the game is connected to the storyline of those first two BioShocks. "I don't want to think about that," he said. "I don't think it's particularly constructive to have that conversation."
"It wasn't right after BioShock that we determined what we were going to do," Levine said. "We did sort of play around with a bunch of different things. One thing that was quickly apparent to us was that BioShock 2 wasn't the right product for us because of when they wanted it, because the company wanted it to be in Rapture, which makes sense, they wanted a follow-up. As our team, independent of BioShock 2, we had said what we wanted to say about that world as a studio."
"We were really looking to take on a project where we had a basic framework for some things — look, we had a framework for BioShock as well from a previous game we did [System Shock] — but also, and I've said this to other people, that there will be no sacred cows. Anything was up for grabs in terms of what would be there and what wouldn't be there."
Levine identifies two qualities that define BioShock Infinite as a BioShock game.
First: "It is set in a place that is both completely strange and fantastical but also strangely grounded and strangely familiar, believable," he said. For the first two BioShocks that location was Rapture a failed underwater city built with Objectivist principles and infused with a 1950s Art Deco design. Infinite's location is the floating city of Columbia, circa 1912, a city populated by proud Americans who view their country as a technologically great, ascendant Caucasian-led God-fearing society. "Fantastical" but "strangely familiar" indeed. Said Levine: "To do another game that felt like that but wasn't called BioShock game would seem dishonest."
Second defining BioShock quality: "The other thing is that when you deal with enemies you have a huge amount of expressability in how you approach a problem set. How I want to play the game is very different from how you're going to play the game." This other quality was a BioShock hallmark. Each enemy encounter in the first two games could be handled with a variety of methods: shooting, hacking, stealth, offensive plasmid (think: magic) powers, deputizing robots, confusing and converting enemies, springing traps, and so on.