Is it normal for Apple Pay to ask for your social security number?

Yes, if you are attempting to ...send and receive money with Apple Pay or use Apple Pay Cash (U.S. only)

"To verify your identity, you may be asked to provide information including your name and address to the bank and their identity verification service provider. This information is used only for fraud prevention and to comply with U.S. financial regulations. Your name and address is securely stored by the partner bank and Apple Payments Inc., but any additional information that you’re asked to provide—such as social security number, date of birth, answers to questions (e.g., confirm street name you have previously lived on), or a copy of your government ID—can’t be read by Apple."