ArtDink’s PS1 games - stuff like Tail of the Sun, Carnage Heart, Aquanaut’s Holiday - are some of the most iconic “weird” games in my opinion. For their time, there was certainly nothing like them; even today, Tail of the Sun is a far cry from any “caveman simulator” like the upcoming Before or the cancelled B.C.
In general, the PS1 felt like the “weirdest” console to me. It wasn’t the first 3D console, but it definitely feels like the first console where 3D games where seriously viable from a technical and commercial standpoint. It feels a lot more open than the N64 by comparison. The PS1’s library is a lot larger, a lot weirder. A lot of really unpolished and experimental stuff, lots of obscure games by even more obscure developers and publishers. Stuff like LSD Dream Emulator, OverBlood, Vib Ribbon, Chaos Break, Intelligent Qube… it feels like a bunch of little companies with small budgets and strange ideas got devkits and tried to make something of them.
I think that sense of weirdness also extends to the visuals of PS1 games, especially that distinct texture warping effect. It makes it feel like the games themselves are bending at the seams.
The PS2 also has a vast library with plenty of weird games, and they have their charm for sure, but the “weird” games on PS2 don’t feel quite as weird as the PS1. There’s more budget, more polish, more formula. I’ve been playing and loving Okage: Shadow King, but it’s definitely not weird in the way of a game like, say, King’s Field or Shadow Tower (though even that franchise felt pretty formulaic after a while).
The closest I get to that level of weirdness these days is stuff on itch.io, or really deep dives on Steam.