Thank you very much for your quick and clear answer.
Unbelievable, but the group to which almost all registered users belong had access to the backend. So did other random subgroups of registered users.
I am the only one who has access to the backend and registration is not possible, I create the users myself.
This is also a very recent problem, a few weeks ago I had a user set a new password via this system and it still worked then.
I can't believe I granted these rights myself, I am a bit security paranoia, some rights even seem absurd and random to me. And the website and rights have not changed for at least two years, except for updates and adding some users.
I also find it hard to believe that support from an extension supplier, who have already been granted temporary access in the past, would have done this.
Backend access is secured via folder with .htaccess password and the only superuser logs in via 2FA. I am changing the .htaccess access immediately and will continue to look. I have some other problems, including errors when I go to PHP 8.0, so I stay on 7.4 for now.
After changing permissions, password resetting works again as it should.
Thanks again.
Didier.