Okay, the problem is reproducible. Whenever I start Firefox now, this happens again.
The "solution" is to wait until the known Unresponsive script warning messages appear. As long as those messages don’t pop up, I can’t access any tab group.
Well, how to trigger these messages? Some time ago it was sufficient to load the tab group list (by right-clicking a tab and hovering over "Move to tab group"), but this doesn’t work anymore (because now it loads all tab groups but without the real tabs inside).
I found out that re-opening a closed tab seems to trigger the warning messages (right-click at tab bar). It doesn’t really open the last closed tab, though, as it seems to open a tab that was closed last in a certain tab group in the previous Firefox session (and not the tabs from the current session).
After re-opening one, two or sometimes three tabs the warning messages appear (until now, at least).
Then I click at "Continue" and after a few seconds all my tab groups are back to normal.
However, it seems that the "open last closed tab" function is broken for the whole session then. Even if you close a tab, the option is greyed-out.