A ban of this sort of thing could easily be enforced. They already prohibit discrimination, and there's already a body of precedent to help adjudicate grey areas. Make it a civil rights violation for landlords to ask for social media handles in applications. Then anyone who wants this kind of anonymity can simply avoid tying their real name to the service.

As social media posts could be used to glean protected information, I don't think it's even be that hard politically to get such a law passed in many states.