Can I copy the entire /var/lib/mysql folder to a different server? (mysql vs mariadb, different versions)

Yes, it is possible. (I figured this out while writing the question)

1. Install MariaDB in the new Arch system, verify that it works.
   Do your changes to the /etc/mysql/my.cnf. E.g. innodb_file_per_table.
   (more of this is out of scope for this question)
2. sudo systemctl stop mysqld.
   (You want to stop on both servers, but in my case the other was already off)
3. Rename the /var/lib/mysql folder, e.g. to /var/lib/_mysql.
4. Copy the old /var/lib/mysql folder from the old Debian (LMDE) system.
5. Restore file ownership to mysql:mysql for everything in /var/lib/mysql:
   sudo chown -R mysql:mysql /var/lib/mysql
6. sudo systemctl start mysqld.
   -> Job for mysqld.service failed. See "systemctl status mysqld.service" and "journalctl -xe" for details.
7. sudo systemctl status mysqld.
   -> This shows that the service is running, but some things are going wrong. This is what mysql_upgrade is for.
8. mysql_upgrade -u root -p
   Notes:
   • It is the mysql root password from the old system!
   • I had to remove one leftover folder from /var/lib/mysql due to invalid database name.
   • The file /var/lib/mysql/mysql_upgrade_info needs to be writable.
   • The process can take a while.
9. sudo systemctl restart mysqld.

CAVEAT: Apparently you cannot downgrade to a lower MySQL version. I tried to migrate my databases to MySQL 5.5 (or rather, the corresponding MariaDB), and the server would not start. I had to install MySQL 5.6 instead. (the MariaDB corresponding to MySQL 5.6 is not available on my Linux distro).