The mitzvah of mezuzah placement on all the rooms of the house is incumbent upon the owner or renter. So the room is not the child’s.
However, Rambam writes: “Minors should be educated to affix a mezuzah to their rooms.”[1] Consequently, many opinions hold that a father can allow children to put mezuzot on their own doors. Since they are of an age to understand and perform the mitzvah, this is part of their training. Even after they become Bar or Bat Mitzvah, they would not have to re-affix it.[2]
Incidentally, if you had your child in mind when making the berachah, he would not make his own berachah.[3]
Nevertheless, some authorities dissent from this view and do not allow a child to place a mezuzah, even on his own room.[4] In deference to these opinions, some authorities rule that if he did place the mezuzah as a child, he should re-affix it after his Bar Mitzvah.[5]
Ideally, an adult should have put the mezuzah on the playroom doorway. However, if a child placed it there, it does not have to be removed.[6] Some authorities suggest that when the mezuzah is taken down for routine checking, it should be replaced by an adult.[7]
[1] Rambam, Hilchos Sefer Torah 5:10.
[2] Chovas HaDar 1:6:25; Agur B’ohalecha 8:8:15-16.
[3] Agur B’ohalecha 7:15.
[4] Sha’arei HaMezuzah 16:4; Mezuzot Beitecha 289:2.
[5] Sha’arei HaMezuzah 16:5.
[6] Agur B’ohalecha 8:22.
[7] Sha’arei HaMezuzah 16:5.