The response "me too" may seem awkward to "I love you", but it isn't really and no, not because only fairly socially awkward people say it. It works as an informal response such that the person is saying "me as well" to the statement "I love you" which has been generalized. So, even though they're saying "me too" they're actually meaning "I love you as well".
From my diligent research on the topic I have determined that substantial number of English-speakers find "me too" a perfectly appropriate response to "I love you", while others think it's ungrammatical. So to some extent, I believe the question of whether it's a good translation depends on what variety of English you speak. In some dialects or sociolects, it might be a fine answer in any circumstance, while in others it would sound awkward in most circumstances, and in others yet would be wrong in any circumstance. So there might be some cases where the translation isn't wrong or bad, it's just that the translator's English is a little different from the English you know (though of course there are also cases where the translator don't know English good or makes a careless mistake).
Personally, if I said "I love you" to someone and they said "me too" I'd know exactly what they meant: "I too love you." I doubt that anyone would actually misunderstand and confuse the meaning for "I love me too". Someone might say that as a joke, but it's not a serious interpretation of the statement. However, in some circumstances, saying "me too" would come across as insincere.
Advice on usage: when your significant other is feeling vulnerable or uncertain and says, "I love you," in a desperate or plaintive tone do not answer with "me too."
I took it as Chidori struggling to say something and blurting out whatever first came to her mind so that Nanoha wouldn't leave the room.
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