Coinbase does provide interest on balances held in USDC. But no, if you withdraw it to your own wallet that does not happen.
FWIW, I specifically chose to say "Coinbase" as my mental model here is that Coinbase is an exchange and Circle issues the stablecoin and maybe Coinbase has a deal to get interest from Circle for all the USDC held by Coinbase and maybe even is given some kind of kickback for deposits and withdrawals, but that doesn't really mean much for the business model of running an exchange: Circle directly provides fiat on-ramp services to companies and could to individuals as well (given that their competitor Paxos supports both kinds of user).
FWIW, my defense of Coinbase in specific is that they are a very large exchange that has achieved enough of a network effect that in a world where only one exchange is used by people in the United States, it probably will be Coinbase. The only other competitors right now are Kraken--which I love but have to admit is quite a bit smaller--and... Binance.US... which I think is much more likely to perish under US regulatory crackdown.
But like, Bittrex wouldn't have been on my list yesterday and now can't be on my list as they decided to shut down in the United States, and I think the argument monero-mxr made--that this is because decentralized exchanges now exist and "long tail" centralized exchanges are somewhat pointless--makes a lot of sense.
But then people did the usual thing of kind of trying to claim decentralized exchanges were meaningless as you still are going to likely want at least one company to deal with USD directly... and that went too far, as the decentralized exchange can still dominate the landscape even if you still need this trivial dumb money pipe on the other side, the same way my ISP doesn't make anywhere need as much money as AOL was hoping to because it got commoditized by the world wide web.