Much of how it works depends on what information they have on the person they are trying to catch. Your license will have you date of birth and your address, your passport will have your citizenship, however if those factors are unknown for the suspect, which is likely the case if you only matched by name, then they are not great at proving you are not the bad guy.
I have no doubt that TSA or HSI will eventually figure out you are not who they are looking for, the real question is, will it happen before you miss your train? And having done so, will they alter the list, so it doesn't keep happening to you?