To be clear, it doesn't actually "install" the game to a brand new directory, instead what it does is setup a .desktop file in ~/.local/share/applications/ that points to the TPT executable, so you can open it from your DE's start menu equivalent. And it also associates .cps and .stm files, plus the ptsave:// url, with TPT.

It does this using the standard xdg commands, so wherever that is configured to put the .desktop file and associations, it will go there.