Two answers to your question:

1) Even if a person does not own the property that he lives in, he is still obligated - mid'Rabbanan - to affix a Mezuzah, if he lives there on a regular basis. So the Mishnah (20b) could be referring to this Rabbinical obligation.

2) A Katan can own property. They can aquire things; they just can't give them away. In other words, they have the Da'as to be Koneh but not to be Makneh. You see this clearly from the Din of Lulav on the first day of Sukkos when the Lulav must belong to you in order to fulfill the Mitzvah. See the Gemar (Sukkah 46) and Shulchan Aruch (658:6) that say that one should not give his Lulav to a Katan on the first day of Sukkos because the Katan can aquire the Lulav but he can't return it. (Besides, a Katan can inherit property from his father - MK)

I think the second answer is better because if the Katan doesn't own the property - in which case the obligation is only Rabbinic - it is not clear that he would be obligated because the obligation of a Katan is never more than a Rabbinic obligation of Chinuch and therefore it would be a case of Trei d'Rabbanan - one Rabbinic law on top of the other - which is not likely. This seems to be implied in the language of the Rambam (Hilchos Mezuzah 5:10) that says that we teach children to make Mezuzos l'Bateihem (for their houses), implying that they are obligated only when they own a house.

A variation of your question on the Mishnah (20b): how can a slave be obligated in Mezuzah? Whatever he aquires his master aquires and therefore he can't own property! Answer (1) above would certainly suffice. Furthermore, a slave also can own property if someone gives it to him on condition that his master has no share in it, in which case his obligation is even mid'Oraysa (if he lives in the house as well).

Kol Tuv,

Yonasan Sigler

This is not a Psak Halachah