Why is Apple Pay asking for my underage daughter's Social Security number?

Does she have her own Apple Cash card? That is only for users who are over 18 and verification including Social Security number may be required for that account.

Verify your identity for Apple Card or Apple Cash - Apple Support

You can use Apple Cash Family for users under 18 and then the verification would be for your account.

Set up and use Apple Cash Family - Apple Support

• Make sure she is using her own Apple Account and that the birthdate is correct. If an incorrect birthdate is listed there that would indicate she is over 18, then it may be possible to add her own Apple Cash card, but the verification after a certain amount of aggregate transactions has been reached would fail and likely the account would become locked.

Update the birth date associated with your Apple Account - Apple Support