They’ve never explicitly told us what Monroe’s other name is. We’re not even completely certain whether Monroe is a first name or a last name, although his parents calling him Monroe certainly seems to suggest it’s his first.
Which means not only do we not know Monroe’s last name, if Rosalee took his name when they got married we now have no idea what her last name is, either!
The whole “Eddie Monroe” thing goes back to the very start of Grimm fandom. There was some kind of behind-the-scenes feature in which his name was referenced as Eddie, but both the writers and Silas have since said that it is not canon, it was never canon…it was something a producer did for some strange reason.
As far as anyone but The Powers That Be (and his parents, I guess) know, he’s just Monroe.