Hi mrc247, thanks for your correction - yes, I meant macOS 12 / Monterey. I won't forget that now.
Overall, the Enpass-Chrome interaction looks like a real tinkering solution. After restarting my MBP, I don't seem to have the problem anymore either.
For a long time under BigSur, however, I always had to unlock the EnPass container first before I started Chrome, otherwise it wouldn't work. It's strange that such a topic cannot be handled for such a long time gets - does not speak for Enpass :-(