Can we say "I have a red car. Neither does Sara." or must we say "I have a red car but Sara doesn't."?
I have read this on a website and they said that the first sentence is incorrect but I don't think the same so I need your help.
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The first sentence is incorrect.
Neither is a negative word, but it can't be used except with another negative constituent. Usually the other negative constituent is marked with nor.
In other words, you can't use neither alone to negate something.
Affirmatives have more options:
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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.*
Along with nor as a conjunction:
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.
If you are not making a negative statement in both cases, then neither is not appropriate.
*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.