United's own website states: "United accepts one item of tennis equipment as checked baggage or carry-on baggage. An item of tennis equipment consists of one tennis racket case containing tennis rackets and balls."

Guess one racket is counted as one carry-on bag. Among airline webpages, only JetBlue very clearly indicates that "single racket = one item":

https://www.jetblue.com/travel/baggage/
A tennis racket may be carried on the aircraft and counts as one carry-on item.

A post in this webpage mentions that the racket itself (or rackets themselves) is counted as an item: https://travel.stackexchange.com/que...bins-in-the-us

All info above suggest that a racket (or a tennis bag with of probably 2-3 rackets) should be taken out from other carry-on bag and be carried on its own. Only then, even if it overs 22 inch or the diagonal limit, it should be fine.