Ok, one case of “the error sits in front of the screen”… At least for a part of my problem.
I used the “impersonate” app for logging in account 2. That produced the error that on Linux account 2 the Nexcloud account from 1 has been added.
So after really logging in with the Nextcloud account 2 during authorization, the right account has been used by the Nextcloud app.
After that, however, I had the problem that I had to log in to Nextcloud account 2 after every single Linux login. To solve that, I had to downgrade to client 3.5.1. Now, everything is working.
I should mention that we are using this PC remotely through Guacamole with XRDP. Before the Nextcloud Client 3.5.1 starts, we have to unlock the keyring. I guess that’s the step the Nextcloud client 3.5.2 has problems with.