What makes a smartcard a "multi-factor cryptographic device", and why does the issuer control its software?

It proves possession of a private key through a cryptographic protocol and only works after a second factor (the PIN) has activated it; because the security depends on the embedded software enforcing this, the IdP or issuer is responsible for that software and controls how it works.

"Multi-factor cryptographic device" is the NIST name for this class of authenticator, which also includes security keys with a PIN or fingerprint:

• Cryptographic: the card does real cryptography (a signature with its private key) instead of handing over a stored value.
• Multi-factor: two factors in one device. The card is "something you have", the PIN is "something you know", and the card itself checks the PIN.
• Hardware-protected key: the private key is generated and kept inside the chip and cannot be exported, so the card cannot be cloned.

Embedded software is what turns the chip into a trustworthy authenticator. It checks the PIN, counts wrong attempts and locks the card after a few, and refuses to export the key. The verifier cannot see any of this; it simply trusts that a signature only appears when the PIN was correct. That trust only holds if the issuer controls the software, typically through certified card operating systems and applets, so a user cannot install code that skips the PIN check.

Go deeper:

• NIST SP 800-63B-4: Multi-factor cryptographic authentication — the authenticator type, its activation and key protection requirements.
• NIST SP 800-63B-4: Exportability — what makes a key non-exportable and why it matters.