So why when someone says something, like "I have solved this problem", someone else shouts "me too, me too", not "I too". "Me" is the subject here, isn't it?
But to answer your question, Mav: in informal use, the objective pronoun often appears anywhere that the subjective 'I' is not clearly located in a preverbal position in the sentence.
Few people would answer 'I too', Mav. 'And I' works, though. Your logic is fine, however-- another reason not to try to use logic alone to organize language without support from actual use (and vice versa).
You can think of "I" as implicit in "I, too" if you'd like, but in this case there would not be much point in it, would there? -- because that's not the form that seems illogical to you.
Your puzzlement centers on "Me, too", not on "I, too". By your logic, "I, too" seems the obviously correct way to say it -- by anyone's logic, we might even say! Yet, here we have something that defie
I feel the present tense in the infinitive clause is a redundant expression. "I am grateful for the opportunity to make your acquaintance" is enough. I'll say simply "I am glad to meet you".