It doesn't show Vincent's name on the ID card after Lamar tested the urine sample of Vincent/Jermone:
The film script states that Doctor Lamar already knows who Vincent/Jermone is. He was already aware of Jerome's true identity, to the point that he's been able to discuss with his own son that there's an in-valid working in the heart of the Gattaca program.
LAMAR: Did I ever tell you about my son, Jerome? He's a big fan of yours. He wants to apply here.
Jerome exits up a long enclosed escalator, realizing that Lamar has known all along.
So the fact that the card doesn't say his name is irrelevant. Doctor Lamar already knows Jerome/Vincent's identity, noting that he was in charge of blood testing to try to find him earlier in the film. There's also every possibility that Vincent's name was included in the "Gattaca Bulletin" that all members of staff were instructed to read during the 'flight path' scene.
I'm pretty sure he picked up on a right-handed/left-handed slip by Vincent at some point, but did not call Vincent out, as Dr. Lamar's son was just like Vincent in that he needed to "beat the odds" to make it into space.
Check out some of the dialogue right before Vincent gets on the flight. Dr. Lamar says:
"For future reference, right-handed men don't hold it with their left. It's just one of those things"
Jerome was right-handed, Vincent left-handed. Dr. Lamar just wanted Vincent to prove to his son essentially that you can beat the odds and make it to space, even if the genetics aren't all that they promised to be.
The ID of Vincent shows up, but Dr. Lamar overrides it with Jerome's passing ID.
I agree to a point what both have said that the doctor knew from the beginning that he was an invalid, but his real name is never brought up since the beginning until he tells Irene when she goes to his house and find both of them.
Even though his brother thinks it’s him because of the picture, he never tells anyone. So it’s either a mistake, or something else edited out of the movie.