It’s a noun. “Un vrai de vrai” un vrai mec, un dur, un authentique, a machoman. I think it’s Pègre slang, so it can sound a bit outdated.

It means “really, I swear it’s true !” It sounds both a bit childish and old-fashioned, but not that much.

For a restaurant it sounds a bit random but your interpretation can be correct. That’s not the weirdest name for a restaurant in Japan I heard of.

For other adjectives you can’t use the same form but it works if you add “chez”:

Je suis crevé de chez crevé. -> I am very tired.

It looks less childish and old-fashioned but more familiar that “vrai de vrai”.