Can’t add any menu items on live server (no item list)

I can’t add any items to menus on live server (can’t even see items to add list – there is list of dropdown "titles" like posts, pages, custom post types etc, but when I expand it – list is empty).

Locally it works. Another weird thing is that if I add items to menu locally and export whole wordpress installation with AIO WP Migration to live server, items are there, added to menu. But I still can’t add any other items (on the live server).

Any ideas what is going on? As I said, I have EXACTLY the same site on live server as I’m having locally (exported with AIO WP Migration), but still can’t add menu items on live server, while locally I can…