The technical term is "oxymoronic snake oil"

Perfectly possible. It's "proprietary" in contrast to "standardized", not in contrast to "open". The Signal protocol isn't a standard from any standards body (eg ISO, or the IETF), so it's proprietary. Its spec is freely accessible, and free to use, but it's still one company's protocol.

Is it possible for something to be open-source and proprietary at the same time?

It is not. Although in this case, it could perhaps be interpreted as "proprietary protocol for (interacting with?) open-source encryption".

"Proprietary" does not necessarily mean "paid" or some other technical term relating to a source license - it just means "our own variety of X".

>it just means "our own variety of X".

I don't buy it. Can you find any instances where it's used like that? According to that definition, you could also say "SpiderMonkey is Mozilla's proprietary javascript engine", which is obviously absurd.

Furthermore, the definitions listed on wiktionary and merriam-webster both seem to incorporate some sort of exclusivity. Just because it's your own variation doesn't make it proprietary.

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.

Multi-licensing comes to mind but I dont believe that characterization fits in this case as the git repos seem to be GPLv3 and AGPLv3.