I've had a sick fascination with licensed games for years, but most of the anime based ones I've played have been "ok", not great, not memorably terrible. The two best that spring to mind are Ghost in the Shell and Gundam Battle Assault 2, both on the PS1.
With a wide selection of unlockable mechs from across many different series, Battle Assault 2 should be an unbalanced treat for anyone who digs Gundam, or just 2D fighters and robots. The visuals are late-playstation great and the mechs have a realistic feeling of bulky weight to them. Though just for a tip if you're playing with a friend- when I say it's unbalanced I mean it. Wing Gundam is so broken in this game it's not even funny. Oh and of course,
so is Big Zam.
Ghost in the Shell however is one of the best action games on the PS1,
period. Tight controls that make strafing, and spider-like go anywhere wall/ceiling climbing, really work on a controller. Beautifully done 32bit 3D graphics and one of the best soundtracks on the system. The only problem with it is that it's rather short. Though extra unlockable anime cutscenes created for the game (and found mainly at chapter starts and endings in the campaign) you can get in the training mode might offer some extended playtime for patient types. I just used a code to unlock them after beating the campaign.
Both games have the original voice actors from the English dubs, though GitS is based more on the manga than anything else. Considering that the Fuchikomas barely appear in the famous 1995 movie. And GitS:Stand Alone Complex, which has a game on the PS2 I've got but not played yet, was years away and used Tachikoma's instead.
PS- I forgot to mention GitS on the PS1 has widescreen support, the only other console game from that generation I can think of at the moment which does this is Goldeneye. Even in the PS2 era that was relatively rare.
PPS- Battle Assault 1 and 3 are both "meh". Unless you're really into Gundam Seed. Then 3 might be worth picking up.