I know this is probably going to get me a down vote, but I've had this idea for a while. I wanted to know if it would work or if it would be an awful failure. Every character is the same person from alternate universes; they are brought together to fight some unknown evil. I'm currently working on other things, so I haven't had much time to think about a plot.
It's certainly possible! In visual media (films, TV, comics) this is made significantly easier than text as you can have visual differentiators to help the audience keep track.
Regardless of your intended medium you're going to have to work out a streamlined way of keeping the different versions distinct that the characters can use to refer to each other - having half the room turn around when someone says the character's name is great for an occasional comedic moment but it's going to get really tedious really fast if it's happening all the time. Ideally you'll want your "naming convention" to be no more cumbersome than referring to someone by an name - a good example of this is in Rick & Morty where the various dimension's Ricks and Mortys are referred to by a short code representing their origin universe e.g. "Rick C137", "Rick J22", or by a short descriptor (usually to tie in to the appearance of the character) e.g. "Cool Rick", "Lizard Morty" etc. This helps keep dialog from being bogged down by delineations between the different versions while keeping it clear to whom the characters are referring.
The best bit is because this is going to be a "problem" that needs solving in the story as well as out of it you can make coming up with the naming convention part of the story:
Steve: Hey Steve!
Steve: Oh man, this is going to get real confusing, real quick! Ok - you're "Big Steve", you're "Little Steve", and you're "Scar Steve"
Little Steve: Hey! Why am I "Little Steve"?!
Big Steve: Are you calling me fat?
We get a lot of these "is it possible to write a story where ..." questions. I can't imagine someone seriously saying, "No, that's impossible". Like, what, you're going to be arrested by the Novel Police?
The serious question is, How do you make it work?
There's the rather obvious problem that the reader won't know who is doing what. If you just say, "Then Bob said ... and Bob replied ... Bob ran out of the room, leaving Bob alone" etc., the story is going to be very confusing. There might be times when such confusion is the point of the story. Like, "Nancy was sitting alone when Bob walked into the room ...", and she has no way to know which Bob, maybe she doesn't even know that there are multiple Bobs. But if there are multiple Bobs in a room talking, presumably they know which of them it is who is speaking or doing whatever at any moment.
I think it likely you would have to distinguish them. Maybe that just means saying "Bob #1", "Bob #2", etc. Or "Short Bob" and "Tall Bob" and "Red-haired Bob" etc.
I've read time travel stories where a person goes back in time and meets himself. Usually the writers solution is to just call them "Old Bob" and "Young Bob" or some such. Similarly for parallel dimensions stories.
Novelty is not a reason to justify bad design.
-Mark Rosewater, head designer of Magic: the Gathering
Just because you can do something doesn't mean you should.
It is possible to write a good and/or successful novel using anything as a premise. But some restrictions make such an endeavor significantly harder than others. This is one of them. And writing a good novel is hard enough to start with - you don't need to make it harder on yourself.
The only reason for giving all your characters the same name that I would consider valid is if you absolutely cannot tell the story you want to tell without it - that it is so integral to the story's premise that without it you have a different story. If that's the case then go for it - but novelty for novelty's sake is paid at the expense of quality.
A very interesting concept. How about this: you number them. You have Bob 1, Bob 2, Bob 3, and so forth. Bob 1 is the main character, or the first one introduced, and every time you add another Bob, you add another number.
If you don't want to do it like that, you can give each alternate universe a number (how you would do that is beyond me). Bob 872 comes from Universe 872, Bob 245 comes from Universe 245, and so forth.
If you don't number them, it will be exceedingly tricky for your reader to keep track of who's who--unless you don't care, and all actions are done by a collective group of Bobs. In that case, numbers and personal identification don't matter, and you can blur them all together.