How Chesapeake became prize money for Shannon's crew

Did HMS Shannon's crew receive prize money for capturing USS Chesapeake, and how did that system work?

Yes. The Admiralty bought Chesapeake into service, and the purchase price was divided among Shannon's crew according to the prize-money rules.

The exact distribution cannot be reconstructed from memory during the stream, but the underlying mechanism is clear. The Admiralty purchased a captured vessel into Royal Navy service and distributed that purchase price among the captors. The shares varied with rank and responsibility, so prize money was not simply an arbitrary reward. Chesapeake's entry into British service therefore meant that HMS Shannon's crew received prize money.