Why are you scared of sharing SSN? In Estonia it’s called Personal Code and it doesn’t have any special status, just your unique digital name. We have left these days behind dozen years ago when banks used asking Personal Code as a security measure.
It's also used for things like tracking of tax liabilities - so if someone has your SSN, with some minor fudging of other data there have been historic issues like claiming tax overpayments and getting checks from the government.
It's incredibly dumb, but it is what it is (mostly still).
At least most of the big players aren't quite as dumb about it as they used to be.