The SMS popup only collects information that is part of Clover's customer database - phone, email, name, etc., and syncs this collected information to the master customer database on Clover. Unfortunately Birthday is NOT a field supported by Clover at this time. This may change in the future as Clover evolves their customer app.
So, how to add a customer's birthday to the marketing database? In the marketing app, click on Enrolled Customer tab, and search for the customer by phone number, name or email and then add their birth date. Consumers themselves can also add their own birthday to the marketing database when they redeem a coupon.