All those settings are saved in their own entry (theme_mod) in the WP database. Every theme with a different name has the same “featurre”. There is no workaround for this. And this is true for any theme’s child themes. Is a hard-wired WP issue that is set in stone.
That said, I think this normally only affects the Menu assignments when used with Weaver Xtreme, especially since Weaver Xteme 5 no longer supports the WP background color.
There might be others, but I ususally only have to reset the menu asssignments.