Hermès has a briefcase without the Kelly closure widely available. That's a classic bag that hasn't gone out of fashion since after the second world war. It's more of a classic than the Birkin.
If you want a highly recognizable Hermès bag that's known to a lot of people as hard to get and as a status symbol to a wider public, then no, Hermès doesn't have that specifically in the men's category, even if as mentioned here already many of the high profile bags are unisex. But that's because it's fashion mainly directed at a female demographic that creates this situation, I can't think of any bag equivalent that was mostly marketed in the men's category.