Not OP, but I am half Asian and worked in various Asian restaurants growing up in the US. When I read the term "basic Asian (brown) sauce" I'd assume some combination of soy sauce, brown sugar, oyster sauce, and chicken/beef broth. With options for garlic, ginger, sesame oil, and rice wine, plus whatever else to tailor towards the dish.

I'd wager that having spent so long in Taiwan you've probably never heard the term because there it would just be "sauce" :P (or less tongue-in-cheek, it's just something that is implicitly and silently understood and recognized rather than having some kind of formal name).

No reply from OP, but I know or know of a few people who do something along these lines. There are a set of ingredients relatively common though clearly not universal across East Asia and I suspect OP is making a pastiche of those. If it were a language you'd call it a pidgin. But it's food so I dunno what you call it.

Though the phrase "White Man's Teriyaki sauce" is running around in my head and yes I get the irony of that.

As someone else living in taiwan a long time, You wouldnt hear asian sauce in asia its just sauce haha 東亞醬 doesn't exist. We dont say american ketchup. But in usa numerous friends say make a quick asian sauce to dip the protein which as comments below explain is a concoction of personal tastes.