In my years living in the United States, I've used my SSN several times, but I've never been required to use my Social Security Card. Is there any process or activity that requires the card that your Social Security Number is printed on?
They can be used for ID. For government purposes various type of cards have points and a certain number of points is needed for various purposes. A social security card can be used as such.
For example, when we go to get a drivers license, you can bring an ATM Card, Passport, Utility Bill, SS Card (as examples) for ID. Each of those has a different value (passport high, utility bill low). They add the values you present and they have to be above a threshold.
The law does not ever require you to show your Social Security card, or even possess a physical card. However, your employer is required, by the IRS, "to get each employee's name and Social Security number (SSN) and to enter them on Form W-2". If you present a physical card, that is sufficient. Otherwise, the employer must verify the number. There does seems to be widespread confusion in HR offices over whether an employee must present a physical card.