Gestapo comparisons aside, you do understand that this is a significant deviation from the status quo in the US, right? The US can't even agree on a unified federal ID; what makes you think we're anywhere close to accepting a world in which the government is a party to civil matters like leases?
(And to be clear: enforcing this kind of introspection absolutely would require an enforcement agency with unprecedented visibility into the paychecks and housing statuses of every single person in this country.)
Illegal immigrants have many methods of getting SSNs. Like, it's not even difficult.
It's not impossible but it'd have to go with cultural changes, and have a dire impact on poor people. Just because the US is rich doesn't mean it's working well, just look at healthcare and how costly and unfair it is.