Tags don’t appear in Quick Edit

I have created several taxonomies with CPT UI. They all appear nicely in the edit window, but they don’t in the Quick Edit. I saw your forum answer to a similar question from 2 years ago. But the mentioned solution is not available in my setup (latest WordPress).
How can I set this to appear in the quick edit?

My biggest issue is that to copy a comma-separated list of tags does only seem to be possible in quick-edit. In the actual edit form, they appear as bubbles and I can’t copy them. It’s quite annoying actually. Not sure why they did this.

Thanks for your help