Mozilla most definitely has exclusive ownership of their property. Ask the people who've tried to repackage Firefox without fully stripping the branding, and get prodded by Mozilla's lawyers.

Here it's a case of semantics, whether you're talking about the open-source software or the full package including services from Mozilla and the Firefox branding. Firefox as a whole is certainly proprietary to Mozilla, even if the code is not.

I suspect the intent re: signal is similar, where the code is indeed open-source, but IIRC signal doesn't network with non-official clients?

Probably best to avoid using the word "proprietary" because of the confusion/connotations, but I don't think it's entirely wrong.