adverbs - Grammaticality of "I have a car, neither does Sara"

Let's get the other uses of neither down for completeness, before focusing on this use, because it does help to think about all the ways it is used.

As a determiner or a pronoun, it rejects all of two or more possibilities.*

"Would you like the chocolate or the coffee cake?" "Neither, thanks."

The EL&U question asked which of two sentences was correct, but neither seemed correct to me.

Along with nor as a conjunction:

That is fit for neither man nor beast!

Now, onto the conjunctive adverb sense you're looking at. Just as the other senses are negating all of a set of possibilities, so as an adverb it means not just that the case is not so, but that it is comparably not so.

I don't have a car, and neither does Sarah.

I don't have a car. Neither does Sarah.

If you are not making a negative statement in both cases, then neither is not appropriate.

I have a car, so does Sarah.

I have a car. Sarah does not.

I don't have a car, but Sarah does.

*While there is a long standing use of neither for more than two items, some say it should only be used for two items, with "none of" or similar for three or more. Certainly, "none of" seems better to me sometimes, but neither seems fine other times.