I like apples. - So do I. I am Russian. - So am I. I have an account on Italki. - So have I. Yesterday I went to the cinema. - So did I. And so on.

"I am too" would be used if it refers to a sentence with to be as the verb. If the statement was, "I am an apple" (which of course doesn't make sense, but for argument's sake), you could respond, "I am too." Of course "me too" would work in this context as well. Because there's no verb to reference, it works with many different statements you can relate to.

Should be "so do I" in formal English, "me too" is ok in informal conversation.