I’m almost positive that Monroe is his first name. I mean, I can see friends and acquaintances, even someone’s wife, calling them by their last name if that’s what they were used to calling them…but even his parents called him Monroe, which would be difficult to explain.
That being said, at this point I almost want them to never tell us. Just let it be an eternal mystery and quirk of the Grimm fandom, that we never learned for certain what Monroe’s other name was or even which name “Monroe” even was. Because he’s been “just Monroe” for so long I’m not sure any possible surname or first name would be satisfying.