I'm not sure I agree with the story outlined on the Gunslinger's page. Who said the Lich was the Gunslinger? It doesn't make sense. Did the developers say that? What makes more sense is that the Lich and the Gunslinger are two separate people, the villain and the hero, respectively, as they are modeled after the villain and the hero in the movie, the Good, the Bad, and the Ugly. What makes more sense is that in the past, the Gunslinger initially failed to slay the Lich, allowing the Lich to grow in power and corrupt the Gungeon. The paradox opened a portal to the past, providing the Gunslinger with a chance to fight again, in the past, ultimately killing the Lich, in the past, and preventing the Gungeon from happening.
He's not killing himself. He's finishing a job left undone.
To believe otherwise is to argue that Blondie, after living in the Gungeon for a number of years, became a powerful Lich, for some reason, and then changed his clothes into those of Angel Eyes.
Another person adding some evidence: If the Lich is the gunslinger in the future, why does he use the gun and bullet that kills the past? The Lich isn't his past and it would be confusing and wordy to explain why but I'm sure you can figure it out. As well as a gunslinger concept wouldn't be uncommon in a game themed around guns, so it isn't far fetched to assume there can be more than one gunslinger type character. The more you take in evidence the more it looks like The Gunslinger and the Lich may have some past sporting rivalry or the Gunslinger is simply killing an evil that no other can kill.
My primary reason for believing that the Gunslinger is the Lich is that he only appears when the Lich is killed using the Paradox. His very appearance is the result of an increasing amount of space-time distortion within the Gungeon (Probably because of repeated use of the Gun That Can Kill the Past), and thus is in and of itself a paradox. And one of the most classic examples of a time paradox would be meeting yourself from another time...
There is also the fact that he starts with the Lich's Eye Bullets. How else would he possess such an item? You could argue he claimed them as a prize in a previous battle against the Lich, but then why does the Lich himself still have them in the battle against him? It only make sense for them both to have them if they're both the same person at different times.
I'd also like to throw out that the Gunslinger's relation to the Lich, regardless of what it is, is probably not a "sporting rivalry." I sincerely doubt that the Gunslinger would unmake the Lich's domain if he enjoyed fighting him. Furthermore, if the Gunslinger is "killing an evil that no other than kill," I'd think that that would support the idea that he is the Lich—it would imply that he knows the Lich's weakness (Possibly what his phylactery truly is—something brought up in the description of Lichy Trigger Finger), which is something he would only know now over any other time if he's been the Lich.
I don't think it necessarily bolsters either of our arguments, but I also think it's worth discussing why the Gunslinger killing the Lich completely undoes the Gungeon's existence. Regardless of whether he's destroying himself, I've never seen any conclusive evidence that the Gungeon cannot exist without a master.
Here's my input on how I think that time would prevent the Gunslinger being the Lich. Note that we can ignore paradoxes as we are literally killing the past as the main storyline, as well as a warning that this will be wordy and possibly hard to follow along.
In order for The Gunslinger to be the Lich and to go back in time to kill the Lich (We know he goes back and not into the future because the Gungeon is undone, meaning that 1. The Lich is the cause of the Gungeon coming into existence, and 2. If you kill the Lich in the future, that would not make the Gungeon disappear.) The Lich must BECOME The Gunslinger in the future, that or the Lich is from the past and traveled into the future (Which we have no evidence that happened or to suppose it happened) This means that we would need a way for the Lich to change from this spectral thing into the Gunslinger (Which you could write off as magic). As well as if the Lich does become the Gunslinger, that means that The Gunslinger must already be in the past (In order for them both to exist at the same time, either the Gunslinger went back in time or the Lich went forwards in time. The only way we know to go back in time is the gun that can kill the past, which in that case he must have already used it, meaning that there's no point for him to be a character in the current Gungeon scenario)
Probably the strongest supporting evidence from this all is that if the Lich is the Gunslinger from the past, that means he would not exist after killing the Lich (Which you COULD possibly write off as he killed the Lich as his past, and that is the reason he has no past, thus writing the left off as the ignoring paradoxes rule) However another point which I cannot think of a way to dispute, is that we have no evidence to believe the Lich went into the future, as if the Gunslinger is traveling through the past, he would not need to travel twice because he could just go straight to the Lich, and wouldn't need to stop halfway to get the Gun and Bullet that can kill the past. Meaning that the ONLY way is for the Lich to travel to the future for them both to be present at the same time. We have nothing to suppose anyone can go into the future. (We can write off that the Lich needed to be in the past to build the Gungeon but he could have just swooped in and taken his place in bullet hell)