Yes and no.
Forget about "either" for a moment, that's not the bigger issue.
In general, you would not use "the car" and "cat" together like that ("the car, cat" <-- notice a space after a comma).
I don't know much about the car or the cat = correct
I don't know much about the car or cat = correct
^^^^^^^.... the car or cat = the (car or cat)
Examples with "either":
1. You and I are talking about horses. I tell you I don't know anything about horses. And you say, what about cars and cats? And I say, "I don't know much about cars or cats either". = I don't know anything about cars, and I don't know anything about cats and in addition to not knowing anything about cars and cats, I also don't know anything about horses.
2a. I don't know anything about cars. Cats either.
2b. I don't know anything about cars, cats either.
2c. I don't know anything about cars -- cats either.
^^^^ all the above = I don't know anything about cars and I don't know anything about cats.
3a. I don't know anything about the car or the cat.
3b. I don't know anything about the car or cat.
^^^^^^^ both mean the *specific* car and the *specific* cat that we were talking about.
4a. I don't know anything about the car or the cat either. = I don't know anything about the car and I don't know anything about the cat.
4b. I don't know anything about the car or cat either. = I don't know anything about (the car and the cat) in addition to something else (x), horses maybe.
Basically, the problem isn't "either", it's ("the car, cat")