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I am noticing almost every game that indicates it is compatible with Linux specifies Ubuntu... they never say any of the other common Linux distros I hear about like Linux Mint.

Will a game that is compatible with Ubuntu pretty much always work with other distros like Linux Mint?
This question / problem has been solved by Themkenimage
A game that is compatible with Ubuntu will pretty much always work with other distros, like Linux Mint, that are derivates of Ubuntu or flavours, like Kubuntu. Debian, which Ubuntu is based on, should usually work too.

GOG only supports those but some game developers try to ensure their games work on other distros too but it is not guaranteed.
And they'll be mostly compatible with many other distroes too.
Save for the occasional library file that doesn't exist. Libclunk for example is MIA but required to run Vangers on Linux.
https://www.gog.com/forum/general/adamhms_linux_mint_beginners_guide