Upvote:2
Essence answers the question "what is this thing?"
A proof for God which reveals something about God's essence is St Thomas' third way. The gist is that everything in existence has its reason for being in something else. In other words, everything we see seems to lack aseity (or, they are contingent). But if everything lacked aseity, then at some point nothing would have existed, and therefore something would have had to have come from nothing. This is absurd. Therefore there has to be something which is has its reason for being in itself, and we refer to this being as God.
Notice that the Trinity or Personhood is not brought up in that proof, but we just showed that God's essence is His existence. This is because we are speaking of being, and as we know God is only 1 being. The Persons don't "share" parts of the Divine Nature; to say otherwise is either a sort of Sabellianism or polytheism. Each of the Persons is God, and not a separate being.
But to say that the Son (or Spirit) has His reason for being in Himself, seems to imply that the Son is a separate being. And so the Persons of the Trinity are described in relation to each other.