Sriracha is the product brand (e.g. Mountain Dew). Huy Fong Foods is the manufacturer brand (e.g. Pepsico). But like kleenex and saran wrap, it has become a generic term. Sriracha is a proper noun, but is slowly becoming a common noun.
It's gotten to the point where there are numerous competitors, some of whom produce bottles that look exactly the same except for two things: a yellow cap instead of green and a slightly different label/print of the logo and other information.
I think packaging falls under the rubric of "trade dress" -- different than trademarks, which generally involves words and symbols.
So if a competitor is selling red hot sauce in a bottle shaped just like a Huy Fong Sriracha bottle, along with similar writing, Huy Fong might have a case for challenging that, regardless of whether the competing product says "Sriracha" on it or displays the symbol of a rooster.
I was at the Fairway up on 132nd and riverside Dr last weekwmd, and they.had the competitor sauce you speak of. The big difference I noticed was they did not have any sodium benzoate preservative like the original siracha. If only they would give up in the preservative, I'd eat it again. Some real good all natural alternatives out there though.