What would be a "vender-neutral" solution in this case? There are very very few people that host their own email servers on the internet meaning almost everyone hitting this login window have a gmail/yahoo/whatever email address.
Most people have jobs with companies that run mailservers. There are even community mailservers. Email is a free protocol, and as such it's vender-neutral by default.
But it makes more sense in the context of a reply further down: "NYU is a private university. The social media login option is only for visitors who are engaged in onsite research. Our materials are not available to the public online." If they are a private company, of course they can do as they please.
Tbh, there are two possible explanations:
1. Someone tried to save a bunch of money and data-handling responsibility.
2. Someone sold tracking data and users to Twitter and Facebook.
Even some of the major "social" providers support this; e.g., I think Google does.