In Russian, I have been told the punctuation is important to let us know when the phrase is actually a sentence, since the conjugation of the verb "to be" is omitted. So "I am too!" is really the best answer when the first letter is capital and there is ending punctuation.
I feel like "me too!" has simply become a fixed phrase for agreement, even if it's not grammatically correct.
"I also" makes more grammatical sense when you replace the implied words ("I am hungry" "What a coincidence! I am also hungry!" not "Me am too hungry!), but it just feels...stilted and not like something anyone would ever say out loud.
"Me, too." is grammatically correct in most instances. For example, if someone said "He rekt us", an you respond "Me too.", you are correct, because you are saying "[He rekt] me too." Saying "I too." would be grammatically incorrect because you are saying "He rekt I too." But however, if someone were to say for example "We are Power Rangers fans.", "I too" would be grammatically correct, because you are saying "I, too, am a Power Rangers fan."
However, "Я" is a subject pronoun (Меня would be the accusative case), while "Me" is an object pronoun. I is the subject pronoun in English. So "I, too" or "I also" would be more correct.
The answer that is accepted is "I am too!" and the key to recognizing the omitted verb is that the first letter of the sentence is capitalized and there is ending punctuation. They are mercifully also accepting the very common English way of putting that which is "Me too!"
While мне would translate to "me", "me too" is technically not correct grammar, even though it has become common use (see comments above).
So "me too" can't be translated literally - you have to go with "I" instead of "me". That goes for the other languages I know, as well (German and Spanish).
Pronouns are the last vestige of the case system in English, and when using this construction in speech, we always use the objective case (me/him/her/us/them) in this construction--even when it's the subject (I/he/she/we/they) of the original statement.
Like, "Joe refuses to eat hamburgers. He's a vegetarian" "Really? Him too?"
"The teachers have joined in the public-sector worker's strike." "Them too?"
Even though Joe and the teachers are subjects, "he too" and "they too" only feel right in a complete sentence--and even then, they sound a little affected (He too refused to eat hamburgers. They too joined the strike).
I wonder if there's a logical explanation to it, or if it's just more of English being English.
I would say it depends on the context of "me too". If you tell your friend your teacher gave you a good grade and your friend says, "Yeah, me too", then in German the last bit would be "mir auch" (ie, "me too", and not "ich auch").
I'm curious what a Russian speaker would say about the use of pronouns in tag sentences!
My guess is: у меня тоже would be if you're saying that you also have something, since outside of English, the common thing to indicate agreement is to mirror the case of what the other person said. (In spoken English, it seems to be "me too!" no matter what.)
Тебе холодно? Мне тоже! (It's cold to you? [i.e., You're cold?] It is to me too!)
У тебя собака? У меня тоже! (You have a dog? Me too!)
Which means you have to pay attention to the case when you're dealing with a transitive verb to know whether you're agreeing with the subject or the object.
Она хочет убивать тебя? (She wants to kill you?)
Меня тоже! Me too! (She also wants to kill me!) vs. Я тоже! I do too! (I also want to kill you!)
English has a strong French tradition. As such "me too" is usually heard. Similar to "moi aussi". Ironically the U.K. has a much stronger knowledge of French than the U.S., but shuns Frenchisms. I would think you might hear "I too." in the U.K., but "me too" would still have more usage.