Me Either and Me Neither have the same meaning when you agree to a negative sentence. They are used differently in other contexts. Me Either is more common in the U.S.A and Me Neither is more common in the U.K and both are used in an informal way.
I don’t like tennis.
Me neither/Me either.
There are differences in the way that we use Either and Neither.
We use Either with negative sentences.
We use Neither with positive sentences.
I don’t like either tennis or golf.
I like neither tennis nor golf.
Notice how we usually use Either. You can’t use “neither” in this sentence.
I don’t like tennis either
I don’t like tennis neither.
Formal and informal English
Me neither and me either are informal English.
Me either can sound incorrect to many people. Many people use it nowadays.
The correct formal way to express agreement is Neither do I.
I don’t like tennis
Neither do I.