There is no privacy-focused PayPal alternative in the US, in part because US money transfer laws and policies (e.g. Know Your Customer) directly oppose privacy.

However, there are a couple of new projects that might eventually lead to something less bad for privacy than PayPal is:

• GNU Taler, if they ever get any exchanges, and they either figure out how to mitigate the high fees for wire transfers or use some other settlement method when people on different exchanges make small payments. (Their plan to use batch wire transfers won’t help until the exchanges get a lot of adoption and frequent use. Of course, high fees discourage adoption and use, so this might not ever happen.)
• FedNow, if banks ever use it to offer appealing person-to-person payment services instead of just using it for themselves and their business customers.