Like a lot of people, the late 90's anime boom where I got exposure to quite a number of anime, though I think my first exposure to anime was an obscure anime adaptation of The Wizard of Oz on a VHS.

bobs

Here's a few that come to mind:

Panzer Bandit (PS1)
Advanced Variable Geo 2 (PS1)
Seirei Senshi Spriggan (PCE/TG16)
Gekibo: Gekisha Boy (PCE/TG16)
Mazin Saga: Mutant Fighter (MD/GEN)
Battle Mania Daiginjou (MD/GEN)
Arcus Odyssey (MD/GEN)
Dharma Dojo (SNES)
Magical Pop'n (SNES)
Hagane: The Final Conflict (SNES)
Asuka 120% Burning Fest Limited (Saturn)
Game Tengoku (Saturn)
Baroque (Saturn)

There's honestly a lot more I could think of, but I don't want to flood this answer with dozens of suggestions.

First one I remember purchasing for myself was an Xbox 360. But the first experience I had with one was a SEGA Genesis.

SpongeBob SquarePants has been my favorite for years. The anime and manga I normally see is broad range of stuff, but I do like Urusei Yatsura, Ranma ½, and Slayers in particular. Really there's actually a lot that I just don't talk about for some reason.

SEGA Genesis
SNES
TurboGrafx-16
Playstation 1
SEGA Saturn

A lot of 90's bias.