I only played Ehrgeiz once. It was in an arcade while I was in Vegas in, like 1998 or something, the details are foggy. I remember there were characters. I believe they where on the screen. And they did fight, after a fashion. So, yes, I suppose that would be considered a fighting game.
I honestly don't think of Smash Bros. as a fighting game because of all the crap that gets connected with fighting games (and since the games are very different from normal fighters). I just saw it on the list so I picked it.
I like nonconventional fighting games. Infact, my favorites are closer to brawlers. Anyone remember the massive 3D stage fighter BioFreaks? It was on N64, if you can beileve that. It had it all, gun arms, blade arms, bullets, distopian future, mutants, the ability to cut off the other player's arms (both of them) and them watch them bleed while they try to fight you with only their weak little kicks and you laugh and snip away their health little by little until finally they hit the floor, soaked through with their own blood. Good times.