It's a bloody silly system. It is supposed to be a bit secret, but you are constantly asked for your social as an identifier, so it is in all sorts of systems completely Inappropriately. Some people keep up the pretence that it is secret, other times you will be asked fr the number out loud when you are standing in the supermarket.
You may be asked for the card itself when opening a bank account by the way, or credit card account. My wife has lst hers (from 20 years ago). She was previously here as a student, so was entitled to one, but nw is on an H4 so is NOT entitled to one. In typical "stupid American" catch 22 the SSA would not replace the lost card, so that meant She coudn't get a credit card from Wells Fargo, even though she had a bank account with them and had joint accounts with me honestly, you couldn't make some of this stuff up...