For a small Non-profit business, if you want to include a Geographical Region as part of your business name, then is it a Legally Enforceable requirement that the business has ties to the region (e.g. a Business address)?
Not necessarily, although one can imagine circumstances where it would.
In general, you can put anything in the name of a non-profit that you want. So, if I want to start the Borneo Burger Society in Portland, Maine, that isn't in and of itself illegal.
Trouble could arise, however, if a name containing a geographic location is used in furtherance of fraud upon someone. So, if you told someone that your burger was an authentic Borneo Burger made from cows born in Borneo because the Borneo Burger Society certified it, when the Borneo Burger Society did not in fact have any connection to Borneo, Indonesia, there might be some manner of actionable fraud involved.
Fraud requires that you communicate a material fact that is not true, knowing it is not true, with the intent that it be relied upon by another, and that person justifiably relies your statement and suffers damages caused by that reliance. A false statement (if it is indeed a statement of a material fact at all) is only the beginning and not the end of the inquiry.
Similarly, you can have a "Champagne Lover's Club" based in Portland, Maine, which isn't even conceivably fraudulent. You aren't purporting to make Champagne in Portland, Maine, when that can only be done in France. You are merely purporting to love Champagne, which can be done anywhere on Earth or even in outer space.
Similarly, you can have a restaurant named "Hong Kong restaurant" in Portland, Maine, which isn't suggesting a location in Hong Kong, China, merely an inspiration from Hong Kong, China, and it would be patently unreasonable for anyone going to the Portland, Maine restaurant to claim that they were deceived into thinking that they were in China.