The short answer is yes, it would be normal to respond with either statement.
As was mentioned already in the comments, the "yes/yeah" parts of your example sentences are simply agreeing with the original speaker. It would become awkward if they'd followed their statement with a question ("I don't like driving, do you?" "Yeah, me neither.") and sounds more like you'd formulated your response before they'd finished speaking, but as it stands it's not awkward and definitely not incorrect.
I know you didn't ask about the difference between responding with "me" or "I" but in formal speech they should both be "I" since you are the subject of your own statement. "Me too" and "me neither" are both fully accepted in conversational speech, though.
Having grown up in England and then moved to the States I will say that I have (on incredibly rare occasions) encountered people who are amused by my "formality" when I say "neither do I" but have never encountered the same with "me neither."
So, while both are fully acceptable statements in both the English speaking countries I have any experience with, the States appears to have a (very, very) slight leaning towards "me neither" in an informal setting.