Q: When someone says, “I don’t like beef,” it’s apparently incorrect to respond, “Me neither,” since “me” is an object, not a subject. But I’ve never heard, “I neither,” only “Me neither” or “Neither do I.”
A: “Me neither” is technically incorrect here, but a lot of people use it idiomatically. In fact, English speakers have been using it since the late 19th century.
The Oxford English Dictionary describes the use of “me neither” for “nor I” (and, we’d add, for “neither do I”) as colloquial—more suited to conversation than to formal English.
The OED says the usage originated in the US, but two of its four citations are from British sources.
The earliest example is from the Feb. 6, 1882, issue of the Marion (OH) Daily Star: “ ‘When I get out I’m not going to tamper with any more proverbs,’ remarked No. 2. ‘Me neither,’ responded No. 1.”
And this is an example from You Can’t Win, a 1926 memoir by Jack Black about his itinerant life of crime: “ ‘I wouldn’t plead guilty to anything if I were you,’ I advised him. ‘Me, neither,’ said his partner.”
The OED’s latest citation is from Sharking, a 1999 novel by the British writer Sophie Stewart: “ ‘I don’t know what to do,’ I said finally. … ‘Me neither,’ said Lucinda.”
Getting back to your question, “Me neither” is an elliptical (or incomplete) version of a longer reply.
When someone says “I don’t like beef,” you can respond with a full sentence if you like. You might say, for example, “I don’t like it either,” “Neither do I like it,” or “Nor do I like it.” But the last two sound stilted.
Then there are various elliptical versions of those responses: “I don’t either,” “Neither do I,” “Nor do I,” and the even more clipped “Nor I.” All of these are technically correct, because “I” is proper as the implied object of an elliptical sentence.
In our own usage, we prefer “I don’t either” or “Neither do I” in conversation. We find “Nor do I” and “Nor I” too formal for speech, though we might use them in writing.
As for “Me neither,” we don’t bat an eye when someone uses it in speech or casual writing. (“Me too” is commonly used in response to positive statements, as we’ve written before on the blog.)
But “I neither” is seldom (if ever) heard in response to a negative statement like “I don’t like beef.” It’s simply not idiomatic—that is, not commonly used by native speakers.
from Grammarphobia http://www.grammarphobia.com/blog/2015/12/me-neither.html