After quite some time I installed xampp again. Everything worked smoothly, only when trying to access phpmyadmin, I was greeted with

New XAMPP security concept:
Access to the requested object is only available from the local network.
This setting can be configured in the file “httpd-xampp.conf”.
If you think this is a server error, please contact the webmaster.
Error 403

It is always very helpful to a) tell people to contact somone they don’t have and b) tell them to edit a file without telling them where it is and what to edit. After some poking around and several strange black magic suggestions, I found a few sources pointing towards what also worked for me. cpighin summed it up the nicest:

Open httpd-xampp.conf in a text editor – for instance by typing

sudo gedit /opt/lampp/etc/extra/httpd-xampp.conf

Then go and find the section »since XAMPP 1.4.3«. There it should say

<Directory “/opt/lampp/phpmyadmin”>
AllowOverride AuthConfig Limit
Order allow,deny
Allow from all
</Directory>

Replace it with

<Directory “/opt/lampp/phpmyadmin”>
AllowOverride AuthConfig Limit
Require all granted
</Directory>

Then restart xampp through

sudo /opt/lampp/lampp restart

and you should be good.

In case it does not solve the problem: I also did

sudo /opt/lampp/lampp security