Yes, it's a clean WordPress installation, used for testing and the theme is still Twenty Fourteen. The Polylang settings are the same as above. Bulgarian is the default language and English is the second language.
Adding and removing a Bulgarian menu from the Customize Theme screen works fine. The problem is only with the English menus - they can't be added from the Customize Theme screen, but only from the Edit menu page.
Removing an English menu from the Customize Theme screen (if it was previously added from the Edit menu page) works, but if the same menu is added both as primary and secondary menu and I remove only one of them (for example the secondary), the other also disappears.
Hope that helps.