I live in California. The car has valid Mexican plates. I am a US citizen and only have the US drivers license.
Will I have a problem crossing it back into Mexico?
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I assume you would be driving across via one of the San Diego border crossings. In my experience you're unlikely to be asked about the car.
Most of the time you can drive through the border without even being stopped - no passport check, no customs check, they rely on you to stop if there's a reason you need to.
There are Guardia Nacional personnel that will flag down some vehicles for further inspection. Usually this is a customs stop where they will search your car - typically they'll open the trunk and look in a bag or two if you have them. Less frequently would they do a full search of the vehicle, but it does happen. They may ask where you're going and how long for, but it's unlikely you'd be asked about the vehicle itself. In over 200 crossings at San Ysidro or Otay Mesa (the two San Diego to Tijuana border crossing points), both in US and Mexico plated vehicles, we've been stopped only a handful of times and have never been asked about the vehicle or who owns it etc. None of vehicles I have owned myself.
Now to be clear - they can ask you, so it would certainly be wiser to have documentation to show them if they do ask, however there's unlikely to be any problem - they're more interested about what is in the car than the car itself.
As an aside, do remember that Mexico is a foreign country (unless you're also a Mexican national) and foreigners should carry a valid passport or travel document when in a foreign country. A US driver license is not a travel document.