Passkeys are unique cert pairs for each site. The site gets the public key, you keep the private to login under your account. The site never stores your private key.

To store them simply, turn off your browsers password/passkey storage. Store them in your password manager along with other sites passwords.

I'm like why is my browser asking to store them? What if I'm using another browser? Why is my password manager fighting with my browser on where to store this passkey?

The answer to all of these questions is “For the exact same reason they do all these same things with passwords”

Think of a passkey as a very, very complex password that is stored on your device (or in a password manager) that you can use to log into websites with without ever having to know what the password is, and it’s never stored on the site you’re logging into, even in a hashed format, so it literally can’t be exposed in a breach.

It’s the exact same technology you use to connect securely to every website you visit, except used in reverse.