Why can't I track the playing time without internet, that should be technically possible?

Clocks, pens, paper have been around for some time ... ;-)

It’s no longer technically possible. At one point in the distant past you could have used a stopwatch, but successive generations found that pressing the button twice was too much effort.

Got a mobile phone? Great.
Got an office program? Awesome.

Now - whenever you start playing a game offline, you start the "stop watch" on your mobile.
When you stop playing, you also stop the stop watch.
Then you transfer the name of the game and the time played into a spreadsheet in your office program.
Whenever you play that game, you repeat that procedure, and add up the playing times.
Do the same for every game you play and want the play times recorded.

"Problem" solved.

Edit: (in case you haven't a mobile phone around - yeah, unlikely, but...)

(Under Win 10)
Start --> Alarms & Clocks --> Stop Watch

Startmenü --> Alarm & Uhr --> Stoppuhr

I seem to remember a lot of RPG games that had a built-in clock. Including Chrono Trigger, Kingdom Hearts notably under them.

If you're using linux, you can use 'time' which will track when it starts stops and how much the program is in use vs say the operating system (OS being loading/saving/data transit etc).

In the future, consider using the GOG Galaxy 2.0 forum for any posts about GOG Galaxy.

I tried the following:
1. Disconnected from the Internet.
2. Launched GOG Galaxy. It noted it's in offline mode.
3. Launched a game and left it running for a little over a minute.
When I exited the game, the playtime wasn't updated like usual. I quit and restarted GOG Galaxy, and then it was updated.

If you really are looking for another application to track playtime, the Radeon drivers can do that. You can look into whether Nvidia can do the same, or other utilities built specifically for that. Maybe even a whole other library management tool, like Playnite.

Why do you even need to track the time you played? Do you keep track of how long you watch tv? Or talk to a friend? Or read a book?
If you don't enjoy the game, simply stop playing it.

Some of the game launchers like Heroic or Lutris might track it offline, though I'm not sure.

Lutris does.

I find it pleasant to see how much I play. Not necessarily to control that, but I like the counter, so I thought there might be a way to count playtime on Gog Galaxy 2.0 without internet. There isn't even a way to port data from Galaxy 2.0 to other programs. Maybe even a trick on how to store the Galaxy 2.0 data externally.

Besides the options previously mentioned, there are quite a few alternatives from what I saw with a quick search on GitHub, although finding the ideal one for your needs might take some time.

You can write some script and run the game by clicking it.

Here are some examples of commands which can be useful to do that.

And the OP could also use something like the Out-File and type arguments (PowerShell and CMD arguments, respectively) to send the information over to a TXT file so he/she can check the information whenever he/she wants.

While I haven't bothered with such myself, I can see where it may be handy to know how much time you spend playing a game in one or multiple sittings, maybe for a review etc.

As some have suggested, a script is the way to go, either creating one yourself or using one someone else created. Take note though, that whatever you use, Galaxy or otherwise, it means having such a program run in the background while playing your game, so a bit of an overhead when it comes to resources, which you may not want, depending on the game.

If I was to bother, I would likely just create a simple AutoIt script, and use ShellExecuteWait and time/date commands, saving to a text file or if you can work it out, directly to an Excel spreadsheet, also using autosum. AutoIt can do all that quite easily.

However there are lots of script options out there, and you could even use a BAT file with commands.

The easiest option would be one of the Game Launcher programs that provide a time recording option.