I’m reviving this thread, because I have the same problem.
I actually assigned a specific mobile menu to my mobile theme through the theme settings. This is only possible when the mobile theme is set as default theme, otherwise the menu settings page for the mobile theme cannot be seen.
But when switching the default theme back to the desktop theme the setting is then not applied if the theme is switched by mobile switcher.
What exactly do you mean with assigning it _directly_ to the the theme? Is there some way to do that in the code? Or did you mean the way I described above (which apparently doesn’t work?)