If someone got arrested for a DUI and gave the police my name, legally what are my rights?

You have absolutely have a case against the DWI/felon. He stole your identify and caused you several types  of harm (suspended license, possible out of pocket losses, possible reputational damage, etc.) and you may absolutely sue him for this. As to the police, you *may* have a case against them--it depends on what is standard procedure in a case like this, whether or not they are supposed to immediately compare the fingerprints to check identify, etc. (If they violated SOP, there may be negligence.)

Your first priority is to clear your name and restore your license. You should retain a lawyer to help you with that; the attorney can also evaluate whether you have a case against the police, and whether a case against the felon is economically worth pursuing. Note that if you do have a case against the police, it typically must be filed quickly (short statute of limiations for claims against the government) and in the proper fashion, so don't delay.