So I got a few:
On PS1 I can recommend Polyphony's Omega Boost, where the same people that make Gran Turismo made a Starfox-Like where you're controlling a giant flying robot.
If you can tolerate tank controls then Brahma Force is a pretty speedy first-person mech game on PS1 I can also recommend. I played it for an hour two years ago, but it felt pretty solid for a tank controlled PS1 game.
Check out the Jumping Flash games, they're that wacky PS1 Aesthetic, but it's still a fun game featuring a giant robot rabbit. Also check out the developer's licensed Ghost in the Shell game, where you play as a Fuchikoma (it was licensing the manga), so I'd also call that a mecha.
Virtual On is a cool arcade mecha fighter, it's one of those Sega Arcade titles that gets forgotten until that one guy at RGG Studio goes down the list of Sega Arcade games to put in the game.
In 2D games I can recommend Cybernator/Assault Suits Valken and Metal Warriors: they're side scrollers but there are some cool mech control things I enjoy like how you have to boost to soften your landing.
Finally: just look at Fromsoft's catalog. They got old Armored Cores, they worked on Another Century's Episode (A crossover mecha game), Metal Wolf Chaos (A game Fromsoft was afraid would offend the USA somehow), and Murakumo Renegade Mech Pursuit (an arcadey chase shooter).
Two other titles that sort of fit but I hesitate to seriously recommend:
Earth Defense Force 4.1 and onwards feature mechs as a deployable unit for the Air Raider, assuming you can get RNG to drop them. Also arachnophobia warning: two of the enemy types in EDF are kaiju-sized spiders.
Robot Alchemic Drive is a really silly game with delightfully below-par voice acting and mecha controls I would compare to QWOP. I've been seeing it a lot lately because one of the main characters is just having the worst day of her life because of the giant robot fights destroying her town.