“It was great seeing you.” “You too.” Why not “Me too”?

Solution 1:

When someone says "it was great seeing you," people often respond, "it was great seeing you too." This can be shortened to just "you too," but the meaning is the same.

Solution 2:

The answer 'you too' is a shortened form of 'It was great seeing you too.' The alternative you are suggesting isn't grammatically correct. In another example: 'I had a great time tonight' the response is 'Me too', not 'You too'.

Solution 3:

This is a rhetorical device known as ellipsis:

ellipsis |iˈlipsis|
noun ( pl. ellipses |-sēz| )
the omission from speech or writing of a word or words that are superfluous or able to be understood from contextual clues.

Now, it may seem a little grand to call an abbreviated greeting a form of rhetoric, but that's what it is.