I'm on openSUSE. We use .rpm files. I dunno if there's kinds beyond rpm and deb, but even if there aren't, software often needs to build for each distro separately anyway, or provide sourcecode for people to build themselves. I'm new to building games for linux, but I've built the prototype for my game as a .x86_64 (Godot's default for building linux games). I don't yet know how portable that is either. That's why I suggested Appimage. That should be able to be built once and run on any linux system.
Still, I'm also new to this. There may be a better way.