I don't know what is legally required or not, but in practical terms over several dozen car crossings at TJ the most I or anyone else in my car have been asked to show is a drivers license. Even that is rare as they usually just ask where you are from and as long as you answer with an honest face they wave you through.

I walked across twice and just showed a drivers license. When you enter Mexico, grab a taxi and tell them you will give them $5 to go to Avenida Revolucion (the main drag). If you don't specify the $5 first, you may get charged $10.