Revealing mistake: When Tommy crumples up the test paper he had written for Andy and throws it into a waste basket the metal container shakes as if he had thrown something heavy. (Telling from the noise it makes the item must have been heavy.) (01:23:30)
Revealing mistake: In the scene where Warden Norton is loading his pistol, as he dumps out the bullets and loads them the primers are dimpled. Unfired bullets would not have the dimples from the gun's hammer (even if they were home made reloads, the primer would have been replaced).
Revealing mistake: When Brooks is carving his name, "dust" seemingly like plaster dust vs wood dust is shown falling on the table top, his shoes etc. When they cut to the view of Brooks looking at his work and closing his knife, there is no "dust" on his hand or knife. That much carving with that much dust falling on the table would have left his knife covered with dust. It was clean.
Chosen answer: He probably only knows the name. He doesn't want to know anymore, so he can pin the blame on Andy should anything go wrong. He no doubt hasn't even considered that Andy might have ID so he can access the accounts after his escape.
Grumpy Scot
Also, Andy indicated that the deposits were night deposits, which were in envelopes. So, the warden may not have known the "Randall Stevens" name.