Well, the weird part was caused by my code, it seems 🙂
This is how it works: ACF is checking the manage_terms capability.
At first I didn’t understand because manage_terms is not listed here:
https://codex.wordpress.org/Roles_and_Capabilities
It’s also not available in the User Role Editor plugin, which I was using to assign capabilities to roles.
But manage_terms is a special capability that is created when you use register_taxonomy() to register a taxonomy:
https://codex.wordpress.org/Function_Reference/register_taxonomy
It is set up as a reference to another capability, in the example it is manage_categories. But in my case I had a custom taxonomy where the manage_terms was set to edit_users for some reason, and that explains it all!
