Selective refresh is only for the items that are selected to cause auto refresh in customizer as explained here:
https://make.wordpress.org/core/2016/02/16/selective-refresh-in-the-customizer/
and here:
https://make.wordpress.org/core/2016/03/22/implementing-selective-refresh-support-for-widgets/
So if the theme developer haven’t included this support then you might not see it refresh automatically on save and publish of menu items.