When is it required to give a Social Security Number to an institution? (USA)

If you do the math you’ll find that there are 999,999,999 possible SSNs. Given that part of the number info includes the geographical area that the card was issued from and given the overlap in years issued, there is no way for all of the 313 million US population to have unique numbers. Where the Social Security Administration runs into problems (and it happens) is when they coincidentally issue the same number to another person with the same name.
You are required to provide your SSN to employers and other legitimate income sources and to the IRS. Beyond that, few entities have a legitimate right to request or use it so just say no.

My understanding is that most institutions cannot require SSN as an identifier. I think this has to do with the fact that illegal immigrants do not have SSNs but are entitled to receive services, go to school, etc.

There are restrictions on when a federal government agency is allowed to ask for your SSN (see Privacy Act of 1974). There are almost no restrictions on when a private company is allowed to ask.

That is excellent, good for your company suspending their work with the testing agency!

While it is true that some people share a SSN, it’s not very common, and it’s pretty much entirely due to error and fraud rather than some fundamental problem with the system.

999 million is more than 313 million, so there are plenty of numbers for everyone alive to have a unique one. It’s as simple as checking a number to see if someone else is currently living and has the number, then skipping that one. If your records of who is alive or who was assigned what number are screwed up, then you can end up issuing an in-use number.

But can they require it?
Payroll? Probably. ID on a surveymonkey poll about the management? Probably no.

Not for SSNs issued since 2011, due to the SSN Randomization program.

Wikipedia notes:

[QUOTE=Wikipedia]
The Social Security Administration does not reuse social security numbers. It has issued over 450 million since the start of the program, and at a use rate of about 5.5 million per year it says it has enough to last several generations without reuse or changing the number of digits.
[/QUOTE]

This is partly as a result of no longer encoding a geographic area into the number.