This isn't a recent statement from TSA.
It is part of the existing TSA regulations I found online.

This part pertains to everyone, not just illegal aliens.

. . . We understand passengers occasionally arrive at the airport without an ID, because of losing them or inadvertently leaving them at home. If this happens to you, it does not necessarily mean you won't be allowed to fly. If you are willing to provide additional information, we have other ways to confirm your identity, like using publicly available databases, so you can get to your destination.

If we clear you through this process, we may give you additional screening. If we can’t verify your identity, you may not be allowed to enter the screening checkpoint or board a plane . . .
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TSA has access to I.C.E., Border Patrol, F.B.I., Etc. databases.
JD2