Northern Ethiopian Amharic (and I believe Tigrean) speaking people in Ethiopia refer to themselves as “Habesha”, which was historically westernized as Abbysinia to refer to the people and country. Since most Bay Area Ethiopian immigrants, if not US in total, are Habesha I don’t think referring to the food as Asbyssinian would provide anything different.
There are some regional variations in foods and staple goods in Ethiopia, but I don’t believe that’s represented in Bay Area restaurants thought the dish names may vary.
My Abyssinia’s menu looks typical of local ethiopian restaurant, and with the exception of Doro wot, which is eaten mostly at holidays, is typical of what I ate in Addis Ababa and Tigray
