As @Satam Sync suggested activation of two factor authentication seems to work.
If you already use it you can try deactivate end reactivate it.
For your interest this is the support answer from Apple I received:
Hello Oliver,
My name is Steffen from Apple Developer Program Support and I am happy
to assist you today.
I understand that you receive an error message stating your session
has expired when trying to login to Xcode. We are aware of the issue
and we are working with priority yo resolve the issue as soon as
possible.
We kindly ask you to try to login again at a later time. We apologize
for any inconvenience this may cause in the meantime. We hope and
thank you for your understanding. As mentioned please try to doing
again after a few hours.
Kind regards,
Steffen Apple Inc.