It's definitely a "simulated" wood grain steering wheel. Plastic really, with sort of a grain carved into it. The grain wears off after many miles. That's normal. Many are cracked, too.
They can be added quite easily to early A's that did not originally have a wood grain wheel. I've got one in my 63 Valiant. No cracks, but the grain is worn off. At a Mopar show a few years ago, one of the judges (?!?!) asked me why I had a brown steering wheel in a black interior. (PS: black was not an available interior color for 63 Valiants, but I didn't know that until fairly recently.)
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The steering wheel was first offered as an option in the 64 model year. Your medallion is correct for 64-65 Valiants and Barracudas. 66 Barracudas used a different center emblem in the horn button. Not sure about 66 Valiants. Darts had their own center emblem.