As an F1 student, you have a restricted social security card.
When your green card comes in the mail, and if you get a conditional green card, do yourself a favor and immediately apply for an unrestricted one.
In a little bit less than two years, you’ll be applying for ROC, which is currently talking anywhere from 1.5-3 years. If you need to change jobs in between, and your extension letter expires, if you don’t have an unrestricted SS card you most likely won’t be able to get a new job. Both of you do have an unrestricted one, together with a valid DL, you’ll be fine as per the I-9.
Planning ahead can save you a headache.