First, to disagree with tye's blanket statement, using "SSN" as a field description is sometimes quite appropriate. For example, I work in the US Payroll industry. Without an SSN (or an IRS ITIN, same format) you are virtually not employable: the employer cannot report your wages to the IRS, nor can they file their own paperwork, and the SSA cannot maintain their wage and earning records. (Remember, "contract" employees aren't really "employees" count for Payroll purposes.) An "ID Number" would be a completely inappropriate replacement for this field.
Short history of the SSN
If you think about it, until recently not everyone needed a SSN. At some point, not everyone needed an SSN: those not eligible (foreign workers depending on status prior to 1986, 1981, 1977, etc..), all of those stay-at-home parents who never worked or filed prior to 1988, any wage earner who chose not to participate prior to 1962, children prior to 1988 (and unclaimed dependants even now), and so on. So the SSA has only issued 140 million SSN's over the past 70 years. Don't get caught in the mental trap of using the SSN against population as an ID number, it doesn't quite work that way. It's not 1:1, and not even 1:N where N is constant over the last 70 years.
They've got plenty left for a while to go. Not another 70 years for sure, but for a while.