I own nearly 200 Linux games on GOG, and I've never had to run any of them as root, neither should you. The only plausible explanation I can think of is that you somehow executed the installer as root.

So you either uninstall it and make sure to install it again as a normal user, or execute this in a terminal:

> sudo chown -R USER.USER DIR

Where USER is your username, and DIR is the path to where the game is installed.

He's saying that the INSTALLERS have to be executed as root. Otherwise they can't be executed.
I can't verify this myself since Puppy Linux is a system best designed with root as the main user, but maybe someone else would find out the bottom of this.

However, what I remember doing on Mint back then was that I gave the installer.sh file execution and read and write permissions and then launched them with a single double click. Maybe that should do it.

I havent had that problem either with the few games i tried installing. I am new to linux so with Windows logic, maybe your user didnt have the privileges to install a program? I am using linux mint 17.3 at the moment.

Well, the OP mentioned start.sh, which is the script to run the games once they're installed.

Exactly, start.sh is no installer. The installers look like gog_*.sh and you have to "Allow executing file as a program" if you want to install them by clicking on them.
Since you have a start.sh file, your game has already been installed (or unpacked which doesn't make a difference) and you don't need to change anything about the file.
If you install the game as root, the start.sh file will require root permissions but it doesn't if the game is simply unpacked or installed the regular way.
Did you possibly install the game as root?

The way that definitely doesn't require any root privileges (either):
cd "directory of choice"
unzip "path/NameOfTheInstaller.sh"

NEVER INSTALL ANY GAMES AS ROOT EXCEPT THOSE FROM YOUR DISTRIBUTIONS REPO. All you need to do is giving the installer "execution" permissions. You can do this as your regular user. "chmod +x file.sh" does the trick. GOG had at least debian archives (and tarballs), but they assumed this was not easy enough for newcomers. *shrugs* shit happens.

One warning: If you mistype this command, you could mess up your entire system.

Hence, I recommend typing everything but the sudo, triple check to make sure the command is correct (in particular, you don't want to run this on anything not in your home directory), and only then put the sudo at the start of the command.

(Also, you shouldn't need to enter this command in the first place unless something has gone really wrong.)

Oh well. Checking back at the OP's post, he mentions installing it, but then says that GOG should allow more distro installation packages. So at this point, He also says that "the installer located at /path/to/installer/start.sh must be run as root" which initially made me think of it as the installer.

In short, I'm confused.

Sound like he installed it as root and as such needs to run the start script as root.

Thanks all for the help, I did install the game as root and that was my problem. I
learned a bit of a lesson from this and now I'm a lot more cautious about installing
things as root.

Is Puppy Linux working OK for you now? Have you fixed the driver issues?

Ah, sudo is your good friend. :-)

--
No, GOG games do NOT need root nor sudo.

Actually I got my HDD now, so I'm not that concerned for Linux. That said, I still have it as a secondary solution in case my PC goes bonkers.

But no, I didn't solve anything. I just left the system as is.

You can root linux with some proper step and carefully.