I did further troubleshooting:
Changing from eth0 to wlan0 did nothing to solve the issue.
I then checked DNS requests in pi-hole and it seems a successful (DNS) request is made after clicking the „Save“ button on the Settings tab.
Time Type Domain Client Status Reply Action 2019-07-15 15:12:46 A www.max2play.com 192.168.XXX.XXX OK (cached) IP (0.2ms) Blacklist 2019-07-15 15:12:46 AAAA www.max2play.com 192.168.XXX.XXX OK (cached) IP (0.2ms) Blacklist
Based on that I can only guess that either, low-level networking is borked in some way (which should be easy to troubleshoot), or, the license activation function somehow fails.