I have asked the host provider to look into the problem. However, I just tried again, and there is no problem with e-mail service from the Joomla-core. It delivers mails to both new users, old users and admin as it should. Other extensions like Event Bookning also uses the e-mail service without any problem.
This, together with that no activity is registrered in the log still makes me belive that the problem is to be found in the GDPR-extension.
I'll get back to you with the response from the hosting provider asap.