since my search for this revealed no result

I can only store items with the controller and not with the keyboard
not to a chest nor to my follower

Just to check, you are using the R button to store things.

Does it work if the controller is disconnected?

Did you remap any keys?

When did this start happening and has it ever worked with the keyboard? If yes to the latter, have you changed the mapping of the keys at all and have you checked to make sure that all the controller options have been disabled in the options screen?