I have read in a book about the theory that explains why in many languages pronouns meaning first person contain letter 'm' (e.g. me, moi, меня, mich) and pronouns describing second person contain letter 't' (e.g. toi, ты). Unfortunately, I can't remember the book nor the theory. Any ideas?
For the specific pronouns that you mention, the explanation is that those languages are Indo-European and have a common historical origin (*me, *tu). This happens to extend to Uralic, and the branches of what might be Altaic (if there is such a family). So there are conjectures that these language groups may also be historically related. This article gives reconstructed pronouns for various language groups. There is a fair amount of controversy over whether the similarities are coincidence, borrowings, or evidence of genetic relation.
There are some very instructive maps on WALS chapter 136 with a lot of food for further thought. Note that besides the M-T-system there is also a N-M-system prevalent in the Americas.