I asked my dentist why and his front-office team said it's because it's easiest to sync with insurance this way. Since I had my insurance card on me, I just used that number and they were quite happy. It just seemed like a convenience thing.

Just say you were born abroad, are not a US citizen etc and therefore don't have an SSN, you just happen to speak good English because you went to American school.

But you're happy to give them your national ID number from your home country. Bonus points if you actually have another passport to waive around while claiming this.

This doesn't work all the time, but it's surprisingly effective in like 90% of cases.