How do they "manage" passkeys? There's nothing to manage except your fingerprint/face authentication.
It really depends on the platform - but in short you'll either need a phone, or be locked into an ecosystem (browser, OS, etc) making using them on multiple devices & browsers difficult or impossible. A password manager supporting passkeys makes this easy as you can 1-click generate a passkey, and 1-click sign-in to services from any device or browser.
And: does using a third-party passkey manager open up passkeys to the same security issues as password managers? Specifically, more than remaining within the Apple-or-Google-supplied system?
Also what security issues with password managers? There's some potential concerns with extension-based over OS based systems, but if your device is compromised where someone can actually access memory then they'd both be equally void to some extent, AFAIK there's nothing seriously concerning security wise on a password manager vs keychain, etc.