Fish store; person who owns or works at a fish store. Thanks to Gigi Simeone. “The fishmongers had suddenly become personable and were saying things like ‘Enjoy!’ as they handed over the fish.” (Nora Ephron, The New Yorker, April 21, 1997)/”Now comes Eataly, an enormous and enormously crowded new Italian-food market and restaurant collection that opened recently off Madison Square Park: 50,000 square feet of restaurants and peninsular provisions, with a fishmonger and butcher (and vegetable butcher) and an espresso bar, a wine store, a cheese store, a cooking school, a kitchenware department and a great deal more.” (Sam Sifton, New York Times, October 20, 2010) Google Ngram.
notoneoffbritishisms.com 
Personally I think “vegetable butcher” is pretentious, although I buy all my meat and vegetables from a supermarket anyway.
I am surprised that fishmonger has made it as a NOOB; Ngram shows US use at about a quarter of British use, but that is still a lot of use. Monger is a common combining form meaning dealer, and is still active. As well as traditional coster- (fruit and vegetables), cheese- and iron-monger we have between us in BrE and AmE generated figurative scare-monger, lawmonger, scandalmonger, warmonger and a fair few others.