If I play fortnite or warzone on e.g. Ps5, am I then automatically with xbox players or do I have to activate crossplay?

But with crossplay you have the pc tryharder with you…

It differs from game to game. In Fortnite you are only in crossplay lobbies if you play with mouse and keyboard or if you have someone in your team who is playing on another platform. In Modern Warfare you can choose whether you want crossplay or not.

Is then playstation and xbox together by default or not

No, switched off with 3 clicks

Does the Ps5 have a crossplay mode that you can play any game with PC?

No they are not. These are two completely different platforms and only a handful of games have a crossplay function.