Go to Food in your dashboard and click on menu groups. Edit the group you’re working with and you can assign the background there. There’s also a spot in the theme options where you can assign some fallback backgrounds. That can be good to have in case a visitor ends up on a page that doesn’t have a background set.
I logged in just now to see if I can spot what is happening and I can’t find it. The menu group images work fine on my development site and the demo site. I even set up a fresh site with the theme to see if it would work there too and it did.
I tried deactivating and reactivating plugins, but that didn’t change anything either. I also tried switching the permalinks back to the plain version. There is a slight chance you might run into trouble using “menu” for the taxonomy slug since “menu” to WordPress refers to navigation menus. When two things share a slug, it can cause unexpected results. That wasn’t the trouble here.
Have you made any changes to the theme? I noticed that updates were disabled when generally is the case when themes have been edited, but not always. If you haven’t, have you tried deleting the theme and installing it again from a fresh copy?