Well, I could be wrong. By "diligent research", I meant "5 minutes of googling", so there's bound to be some inaccuracies in my method.
Lots of stuff on this “me too” construction in various places:
https://www.grammarphobia.com/blog/2010/08/me-too-i-too.html
“English speakers generally choose “me” over “I” when a pronoun is the subject of an elliptical, verbless sentence, never mind what’s technically correct.”
There’s a scene in The Godfather where Michael talks to Kay on the phone with a bunch of wiseguys hanging around, and although he doesn’t say, “Me too,” it’s exactly the situation where a bashful person would say it:
KAY'S VOICE (over the phone): I love you.
MICHAEL (into the phone): Huh?
KAY'S VOICE (over the phone, louder): I love you. (then) Michael?
MICHAEL (into the phone): Yeah, I know.
KAY'S VOICE (over the phone): Tell me you love me...
MICHAEL (into the phone): I can't talk...
KAY'S VOICE (over the phone): Can't you say it?
MICHAEL (into the phone): Eh...I'll see you tonight.
KAY'S VOICE: Okay
[Michael hangs up the phone.]
CLEMENZA: Hey, Mikey, why don't you tell that nice girl you love her? (in an exaggerated Italian accent) I love you with all-a my heart! If I don't see you again soon, I'm a-gonna die! (laughs)