There is no central database of birth certificates in the US and most state, county and city birth certificate records are not online. However, they can be validate if you have one in hand.
It is now the responsibility of the detainee to show proof of legal residency. Prior to Trump, it was ICE's job to determine if you are here legally. ICE's job now is to apprehend those suspected of being in the country illegally and hold them for deportation. There will be a deportation hearing for the group scheduled to be deportation. At that hearing, the judge ask is there any proof showing legal residency in the US. If you have none, you are deported. Fast, efficient, and very prone to error. You may still request a hearing, however the chance of getting a hearing is pretty remote.
What is being done is contrary to our immigration laws and the constitution. The court is allowing Trump to do this because of he has declared National emergency at the border. Challenges to constitutionality is defended using various enemy alien acts that go back as for as 200 years.
If you look illegal, you better have proof on you.