Toss old appointment cards.
They aren’t necessary, although I think some people see them as a reference for past appointments. Many doctor’s offices have online patient portals through which you can look up information; in fact, you might not get an appointment card at all, instead you’ll get a text or email.
If you don’t have access to a patient portal, then an appointment card may seem like proof that you have an appointment. If you enjoy this reassurance, remember to toss past card. Years are rarely noted, and a stack of old appointment cards can be confusing.