This is a late answer, but I'll try to help in case you're still having this problem or if someone else stumbles upon this page. :) Note that this might not be a recommended or most optimal way of solving these kinds of issues, but it's how I do it.
My assumption is that it has to do with Elementary being 64-bit
This can indeed be the case. You can enable the installation of 32-bit versions of packages by running this command in the terminal:
sudo dpkg --add-architecture i386
This needs to be enabled for the following to work.
So, next I'll show you GOG's version of Crimsonland as an example. It did not launch right away, so I ran it through the terminal. You can launch a terminal in the game's folder by navigating to it in Files (Home -> GOG Games -> Gamename), right-clicking the file area and choosing "Open with an application -> Terminal". Then run the command
./start.sh as in the following image. Most, if not all, Linux GOG games are run via the file start.sh
Notice that the error message shows the name of a library: libopenal.so.1
Now you need to figure out what package this might relate to. I searched for libopenal by using the Synaptic Package Manager application available on the elementary AppCentre. I found a package called libopenal1 of which the 64-bit version was already installed. I decided to try installing the 32-bit version of it as well.
I closed the Synaptic Package Manager and issued the following command on the terminal:
sudo apt install libopenal1:i386
Notice the important part: I put :i386 at the end of the package name. This means that I am asking apt to install the 32-bit version of the package.
I proceeded with the installation and afterwards tried starting the game again. It worked!