In my home we have a number of PCs, all running Ubuntu (except one which runs Mint).
Is there a way to automatically synchronize the settings and home-directories of the computers. That way any member of my family could login to either box with the exact same set of credentials.
Im no expert but for your set up it sounds like you will need a central "server". Distributed syncing will not work as all computers may not be on at once. If you had one computer which was on all the time and had a large amount of storage you would have a number of options.
You could mount the users /home folder over the network.
A better option would be to set up rsync scripts that incrementally sync:
You may also have to look at an app called Unison in the software centre but ive never used it and I think it could be quite old.
you can export for example /etc folders on all machines via NFS
then mount those exports on a machine which stays mostly up.
create a cron job, say every 5 minutes to sync back and forth all /etc/shadow files
I assume all the users have the same ID numbers/names on all the machines