No, "either" isn't really a valid answer to "Do you prefer A or B?". Someone might, however, say "I'll take either" or "I'll have either". It's possible some people might say "Either", but it doesn't really fit.
"Neither" doesn't mean they don't like either of them. It could mean that, but it could equally well mean that they love both and can't choose between them. It simply means "no preference". You'd have to judge by their tone of voice or by what else they said (if they expanded on their remarks).
In answer to "I like that shirt", "Me either" is ungrammatical. It would have to be "Me too" (informal) or "So do I" (more formal).
In answer to "I don't like that colour", "Me neither" or "Neither do I" or "Nor do I" can be used to agree with the sentiment and express the fact that you don't like it, either.