Taxonomy field loses data/don't save data

What "ones" are you referring to when you say "one of the two problematic ones"?

Was referring to (Select list, checkbox/radio).

Are any errors logged by Drupal?

Just in case this is due to a browser issue, and it is not actually posting the data:

• Does the browser console log anything?
• Are there any browser plugins which could interfere? Can you try this without browser plugins?

To separate other factors from that particular site, can you reproduce this on a cleanly-installed Drupal site? https://simplytest.me is useful for that.

Can´t reproduce on clean install.

Let's not close it. Instead we need to find out the true steps to reproduce it.