Yes - logically Mark is absolutely right, but people do use "me too" to mean I'll say the same about you
. Mark, if you say to your wife I love you
and she replies Me too
- do you really understand that she loves herself? (And I very much hope you don't destroy the moment with a lecture on grammatical logic!)
A funny story - I had an aunt who said "me too" all the time. At one point there was a whole tribe of kids between 0 and 3 in the family, and she was chief babysitter. She said it so often that one of them interpreted it as being her name and started referring to her as Me too
. It stuck, and the whole family called her Me too
from then on.