^There's a number of ideological issues with Devil May Cry if it isn't taken as symbolism (which it was clearly meant to be).
The Vampire-versus-vampire thing is similar, of simply rejecting one's evil heritage, even to the point of warfare with relatives, those back in one's native culture, and so on. There's no way a demon could turn back into an angel (or rebel against the devil), but that's the thought. It also symbolizes fighting off one's own sinful inclinations while, again, fighting everyone else who didn't make the conversion.
The other thing is, there's no way something TRULY "evil" can fight against evil, HOWEVER, again, this is understood to be symbolic unless someone foolishly thinks in terms of "lesser of two evils" (like violence is evil no matter what so they're doing something "right" but should still be punished).
Dante also isn't supposed to be seen as a morally perfect "hero" in particular, either, except in the sense that he's struggling in the right direction even if he has his flaws.