You're right.
In fact, social security numbers used to be extremely easy to get.
For example, one day in the early 1960's, my father (a lawyer) deposed some money in a savings account in a local bank. The account was jointly owned by himself and our family pet, a doberman named Fritz. The money drew interest, and the amount was reported to the government.
A few weeks later, dad received a letter from the IRS demanding that Fritz immediately get a social security card, and explain why he didn't already have one.
The local farmers loved that letter. It was posted on the cafe (and coffee stop) wall for years.
Was a picture of the pooch pinned next to it? lol