First of all, many thanks for all answers on my matter so far!
I am also pretty new to linux myself, so don't expect an advanced expertise about linux from me, but of course I want to provide more information hoping this helps my helpers to help me run the game ;-)
I downloaded the game the first time and just today, therefore I chose the full one, not just an update. Unfortunately you were wrong, I am using Linux Mint 21.1 with Cinnamon 5.6.8, which is based on Ubuntu afaik.
Something I witnessed, some ren'py games after extracted to their chosen directory come with already set permission to run the .sh file as application - some don't. Why? Maybe this is an useful hint for those who know linux more than I?
I thought okay, just set this permission in the .sh file properties access tab and there we go, but seems as it is not that easy
Guide me to get the information you need and I will provide it if this was still too less information, I will be very happy and thankful if this can be solved. Thanks for all your efforts!