Co-operation is actually a term within game theory and thus used by several board game fans. It is mainly that such things are enforced by the game rules (in risk, monopoly, settlers of catan... we can make a pact but either of us can break it without consequence inside the game).
There are then plenty of multi party games where some measure of informal co-operation is helpful though, Carcassonne being a good example; you are never going to build the good stuff by yourself so while you do not technically have to be nice to other people you kind of do.
Assuming you care more for that than you vs everybody else vs the world and everybody else is also vs the world and everybody else then we do have some options. I assume you also want things to happen in this game rather than basically having people play solitaire at the same time (there are such games).
The simple round a table/party games without the need for things probably starts with mafia/werewolf. Wikipedia I know but it covers it fairly well
https://en.wikipedia.org/wiki/Mafia_(party_game)
If you are actually going into games then there are a few. The earliest that most people will really know is the Dune board game, however it is rare as you like these days and some would probably want to call it a game that features partnerships rather than a cooperative game. A lot of games based on TV shows also seem to have this sort of thing, usually someone is a mole of some form (if you have since learned mafia/werewolf from above then a lot of these play like an enhanced version of that or that meets some popular board game you probably already know) and you need to work around that.
Another example might be pandemic, I have not played this nearly enough to properly form an opinion though.
If you want one that is pure cooperation, similar to how some might ask for a game without any randomness, you probably are better off looking at pen and paper RPGs. It goes way beyond dungeons and dragons too, though before you go too far into that then maybe some of the horror based games might be more your thing.
Anyway
https://boardgamegeek.com/boardgamemechanic/2023/co-operative-play will possibly get you started.
Short version. There is a lot out there if you go looking.
