I can see them in the browser-version and if I got into the store it says that the games are owned, but when I go into my actual games library in Galaxy 2.0, they're not there.

Galaxy 2.0 takes a while to update the games list. It happens to everyone.

You can try to force a refresh by
a) Restarting Galaxy; or
b) Clicking Sync Integrations (in the gear settings menu).

If these don't work, the games should show up tomorrow.. The last sentence is not the most helpful if you want to play now, but GOG Galaxy wants to instil patience in everyone.

Can you download the installers / install the games and play, and then add the games later to Galaxy? (I don't know much about Galaxy, I don't use it)

login via browser. hit:
https://www.gog.com/user/refresh

Sometimes, the library in GOG Galaxy 2.0 might not immediately sync with your account. Ensure that the filters or sorting options in your GOG Galaxy 2.0 library are correctly set. Close and reopen GOG Galaxy 2.0. Sometimes, restarting the application can resolve syncing issues and refresh the displayed library. Double-check your GOG account to confirm that the games are indeed purchased and owned. In GOG Galaxy 2.0, you can try clearing the cache and data related to your account.
I hope you get solution.