How can a Muslim know that it was his own supplication that Allah answered, and not that of another person? And is it considered desirable to ask children to supplicate and for them to do so frequently, based on their innocence?

A Muslim's supplication (du'a) is answered unless it is hindered by an impediment, such as consuming unlawful things (haram). This is not exclusive to children. The forms of answered prayers are varied, as the Prophet, peace and blessings be upon him, said: "There is no Muslim who supplicates with a supplication in which there is no sin or cutting off of kinship ties, but Allah will give him one of three things: either He will hasten his supplication for him, or He will store it up for him in the Hereafter, or He will avert from him an evil equivalent to it." The answering of supplications in these ways is part of the unseen (ghayb) and cannot be known.