I'm not arguing with that at all! I'm saying that the fact that they aren't hasn't actually stopped anyone from using them that way.
When this assumption is violated, a psuedo-identifier is assigned. I have worked with multiple systems that work this way and it is the norm in finance and was the norm in higher education until relatively recently. State government information systems also use SSN as primary identifier surprisingly often, and in the system I've worked most closely with reused SSNs were resolved by just deleting the old record.
You could say that SSNs are the perfect storm - they aren't unique, but they're pretty close to unique, and this allows you to rely on SSNs as an identifier for long enough that when you run into a problem you're too dug in and so you find a workaround. For example, assign non-citizens "SSNs" with prefixes not used by SSA---this was the norm in higher ed, and many institutions "eliminated" the use of SSNs by just handling all students as international.
The IRS, of all organizations, uses SSN as a primary means of identifying individuals. They absolutely run into all kinds of problems with this that must be resolved by using other information as well but that doesn't stop them because it's just too convenient. Many such defects in use of SSNs are resolved by assigning an ITIN, which is just a pseudo-SSN with first digit '9' which is not in use by SSA. Even EINs have the same digit length as SSNs, suggesting that they're handled as SSNs by some systems.
Or consider this similar situation: name and DOB are not unique, but this doesn't stop them being near universally used as primary identifier in healthcare information systems. Actually healthcare widely used SSN before HIPPA mandated a change.