is the flag a brand ?

Answer 1

Yes, I suppose you are right the Flag is the brand for that nation, unit, or cause.

This site covers the US Federal Flag Code: http://www.usflag.org/uscode36.html#173

Unlike a normal brand or logo which can be placed on almost anything their are specific rules on where the flag can be used.

Answer 2

Good question. Its a matter of patrimony, each country reveres its flag as a symbol, an icon. No other country can use it. With all due respect, rather like Coca Cola.