Me too, You too

[joham]

If we hear:
--Glad to see you again.
can we answer 'Me, too' or 'You, too'?

And I once read the following dialogue in a language book written by some Chinese teacher:

--Nice meeting you.
--Me, too.
I doubt whether 'Me too' can be used here. I would have answered 'Nice meeting you too.' or simply 'You too'.

Could you please help me with these questions? Thank you in advance.

[Anglika]

You could say "Me too" in answer to that greeting - it is a normal colloquial use.

In the dialogue I would say "Nice meeting you too" would be more usual.

[RonBee]

I agree with Anglika. The responder could also say, "Same here." The responder could also say, "It's a pleasure seeing you again" (if it's a second meeting).