Each other

No, the pattern is this:

each other

is equivalent to:

and

The sentence "Each other wants a car" does not follow the pattern " each other", and so the "each other" is meaningless. Even if you turned it around, to get "A car is wanted by each other", you'd still only have ONE thing so it STILL doesn't follow the pattern. The essential point about this is that the "things" are simultaneously the subject and the object of the verb - if one is the subject, then the other is the object, and vice versa.

"We want each other" is fine, providing that "we" stands for precisely two people. (For more than two, use "one another" instead of "each other").

But I think that for the car, you'd have to say "They both want a car". In this sentence, the people who want the car are never the object of any verb, and so "each other" doesn't apply.

(For more than two people wanting a car, say "They all want a car").

Rommie