Background checks have been standard in the US for years. The 'application fee' or 'administration fee' charged by the management firm usually covers the standard 'credit and criminal' check. All that will be spelled out in the application that a prospective tenant signs.
Applicants for any federally subsidized housing (public housing, HUD Section 8, USDA RD515, USDA RD514, LIHTC etc.) also have to show proof of legal presence in the US.
My husband had to document residence and legal presence in VA when he got a driving license in 2000. To work legally in the US you provide a social security number; for non-citizens, authorization for work is issued by the Department of Homeland Security.
The system doesn't work perfectly as we all know, but you have to start somewhere.