I'm asking myself how they have done it, or if anyone could explain it to me.
I have found this source where it is described, but I don't quiet get it.
For my understanding, it is not storing the private key on the device, rather on the serves it sends it to. This is apparently done while encrypting the private key with a Yubikey specific key. On authentication, the “key handle” is sent back and decrypted, and signing the challenge with some metadata and sending everything back.
During credential registration, a new key pair is randomly generated by the YubiKey, unique to the new credential. The private key, along with some metadata about the credential, is encrypted using authenticated encryption with a master key. This master key is unique per YubiKey, generated by the device itself upon first startup, and never leaves the YubiKey in any form. For FIDO2 capable YubiKeys, this master key is re-generated if FIDO2 RESET is invoked, thereby invalidating any previously created credentials.
The encryption used for each credential is AES-256 in CCM mode, which allows us to cryptographically tie things like the AppID to the private key, ensuring that the credential can only ever be used with the correct RP. The encrypted (and authenticated) data then forms the 64-byte key handle, which is sent to the server as part of the registration flow, to be stored by the RP for later.
For authentication, the RP returns the key handle to the YubiKey. Here it is decrypted to re-form the private key which is needed to sign the challenge to complete the authentication. Due to the authenticated encryption used, we know that the private data has not been altered in any way, and can verify that the credential is being used with the correct AppID.
By using this approach, the YubiKey does not need to store any per-credential data, and can thus register and use any number of credentials. This is true for both U2F and for WebAuthn "non-resident keys". For WebAuthn resident keys, internal storage must still be used.