Vote #1 Danya Rose, who as far as I know has the right answer.
To my astonishment, OED does not have the phrase. It does have related phrases under cop, v. 3: “to capture, catch, lay hold of”:
- to cop it “to be punished, to get into trouble”
- to cop a packet [no definition]
- to cop a plea “to plead guilty, usually as part of a bargain or agreement with the prosecution”
- to cop a feel “to fondle someone in a sexual manner”
However, under serve, n. 2 we do find the almost identical phase to give (someone) a serve: “to deal roughly wit; to criticize or reprimand sharply”, described as Australian slang. The noun is derived from to serve, and there are three definitions given: “service, adoration” (Middle English); tennis service; and “a serving or helping of food” (“3 serves of the bacon”).
I don’t know enough about tennis to know why cop a serve wouldn’t be referencing tennis. OED does strongly suggest that’s the only way to interpret it..