can you open up separate savings accounts on a shared family Apple Card?

Co-owners must meet credit qualifications and they become responsible, just like the other co-owner, for all aspects of the Apple Card account.

The other option is Participant (Authorized User). They have no requirements in regard to the account. They are not responsible for any aspect of the account. There are no credit qualifications, just age etc. mentioned in the article I linked to below.

Set up Apple Card Family and add a co-owner and participants - Apple Support

No, only owners and co-owners of Apple Card Mastercard accounts are eligible to open an Apple Savings account.