Polylang Some themes don’t use the orignal menu walker provided by WordPress but a custom menu walker to display navigation menus. It is quite frequent that theme developpers forget that plugins can add images in menu items and thus create custom walkers which do not interpret the html code used by Polylang to display flags.
A quick fix is to replace the flags by the language names in the language switcher options. After that, please ask to your theme support how to solve this.