Roughly speaking, you must be:

a/ an American citizen,

b/ a non-American citizen in the US and legally entitled to work,

or

c/ a non-American citizen in the US legally, and not legally entitled to work but able to prove a branch of the US federal government requires that you have a SSN (eg. to receive social security benefits).

Once you have satisfied either of these and obtained your SSN, it will NOT be taken away if/when you no longer satisfy a, b, or c.

As someone who is in the US but neither American nor entitled to work here (I live in the US but work in Canada by the way), I have personal experience.

In fact, recently the state of Georgia enacted a law whereby any applications for or changes to a driver's license have to be accompanied by a SSN. I got my GA driver's license prior to the rule, but now I cannot implement a simple address change to my driver's license without a SSN (or a letter from the Social Security office indicating I am not legally entitled to a social security card, which, as someone who does not satisfy a/b/c above, I am not). (The body that administers driver's licenses is part of the state government, not federal, so I don't satisfy (c) above.

Getting a letter from the social security office indicating I am not qualified to receive a SSN would take a whole friggin' day in line, which I can't spare, so I'm walking around with the wrong address on my driver's license. (Why my address has been changed even though I haven't moved is an entirely different story involving neither federal nor state government, but rather the Atlanta city government.)

All this to say, (a) the US can be incredibly bureaucratic!! and (b) not anyone can get a SSN.