I get the joke but it goes the other way around. Martini is a common family name from the north of Italy (Carmelo is common given name in the south). The drink was named after the name of company producing it (actually half of it.) The company was named Martini because that was the family name of the founder.
The drink https://en.wikipedia.org/wiki/Martini_(vermouth)
The company https://en.wikipedia.org/wiki/Martini_%26_Rossi