Frank knows who Morgenstern is, even though Morgenstern left County years before Frank started working there. Before working at County Frank was a Chicago police officer. He could have had interactions with Morgenstern as a police officer.
Dr. Morgenstern is telling others about Dr. Kostin's history, including saying that Dr. Kostin designed the trauma rooms. By not saying "the original trauma rooms," he is implying the current trauma rooms. But years after the original ER had been built, the ER had been completely gutted and renovated. Once because of an explosion, once as a security renovation to keep people out, unless let in by staff. Even if Dr. Morgenstern didn't know about the renovations, he would have seen that it is not the same ER setup.