It's not a man-purse, it's a satchel. Indiana Jones wears one.

...I keep seeing men with "man-purses". 

Yup.

I even see them for sale in men's clothing stores.  Is that like, a thing here now?  French men do dress in a manner that an American would probably describe as "fancy" (leather loafers, tailored jeans, pink shirts, scarves, etc) but isn't the man-purse pushing it a bit?  Maybe it's just me.

Actually, the man-purses look pretty much exactly like this.