It works for my PS4 console but not for PS5. It doesn't matter if it is a PS4 game or a PS5 game when I buy it on my PS5 console but on my PS4 it works as it is supposed to. Is there some way to make it work for my PS5??

You may have to boot the game once on the console before it shows in Galaxy. (it may not work for all titles, especially new/obscure ones)

I don´t think PS5 is supported yet on Galaxy. Neither is PS3 (my games don´t show up).

I have tried to start the games, restart my console, start some games, turn the console off completely and start the games again. It hasn't helped at all, Galaxy doesn't show my games history on my PS5 no matter what games I have started on it. I will try to reinstall them and se if it works out

Most of my PS5 games have been detected by Galaxy, such as: Ratchet & Clank: Rift Apart, Judgement (has 2 entries, one for PS4 and another for PS5), Nioh Collection, etc.
I can see in Galaxy most of my PS4/PS5 games that I have booted once, but some are missing: Gravity Rush Remastered PS4, Tony Hawk's Pro Skater 1 + 2 PS5.

[edit] nevermind, Tony Hawk's Pro Skater 1 + 2 PS4/PS5 does show in Galaxy.

I don´t have a PS5, but I know PS3 games do not appear on Galaxy.

The Plugin detects PS4 and PS5 games for me. They need to appear in the game library at my PlayStation Network accout. If there are versions for both consoles in my account, I get two entries in the Galaxy library for that game.

That said, the unofficial Plugin is sadly not supported anymore. After recent changes on the PSN API side, you only get games added to your library, but no trophies, no game time etc. will be updated. So it's really barebones. I think, Gog should definitely step their support for official integrations up. It's more then 3 years now and we only got official support for Epic and Xbox...

Sadly, GOG can't officially support a platform without the consent of each company. If they have no interest, there's nothing to be done other than relying on community integrations.

I'm aware of that and GoG was certainly too, as they launched 2.0. The thing is, they promised to unify PC game libraries under a single launcher/library manager and now can't deliver on their promises. This is just disappointing.