Partly it has to do with the highly ritualized nature of this specific exchange: when someone says, “I love you,” anything other than some version of “I love you too” is going to be awkward to some extent. What, “I’m glad”? “That’s nice”? “Oh, aren’t you sweet?” (If you really want to get into regional idioms, try responding, “Why bless your heart!”)
But again, it’s primarily an idiomatic issue rather than a grammatical one. As one hypothetical parallel example, if someone were to say something like, “I’m so glad I met you,” the response, “Me too” would clearly be intended to say, “I too am glad that I met you” rather than “I too am glad that you met me.”
In both cases, the “me too” indicates reciprocity, whatever the strictest parsing of grammar and logic would say.