Praise be to Allah.
I put this question to our shaykh, ‘Abd ar-Rahmaan al-Barraak (may Allah preserve him),
and he said that so long as what he is getting from your father is sufficient, then he is regarded as being independent of means as a result of that, so it is not permissible to give him zakaah.
But if his father cannot afford to spend on him, or refuses to do so, or if what he gives him is not sufficient to meet his needs, then in that case it is permissible to give zakaah to him.
And Allah knows best.