As your question implies, Narnia is a Christian allegory, where Aslan is a stand-in for Christ, and his father, the Emperor, represents God the Father.
With that in mind it's fairly common and orthodox in the Christian church to view Christ as the human face of God, the aspect of the divine that we can understand and interact with. God the Father may seem remote and unknowable, but is considered to be real and present through Christ, and the same can be said of the Emperor through Aslan.
Compare this passage from the Gospel of John ("Chapter 14"):
8 Philip said, “Lord, show us the Father and that will be enough for us.”
9 Jesus answered: “Don’t you know me, Philip, even after I have been among you such a long time? Anyone who has seen me has seen the Father. How can you say, ‘Show us the Father’? 10 Don’t you believe that I am in the Father, and that the Father is in me?..."
With appropriate substitutions, this could easily be dialog from the Narnia series.
With reference to the specific question of why Aslan is the one who sings Narnia into existence, it's a clear reference to John 1:
In the beginning was the Word, and the Word was with God, and the Word was God. 2 He was with God in the beginning. 3 Through him all things were made; without him nothing was made that has been made.
14 The Word became flesh and made his dwelling among us.
The Word is identified with Christ, and therefore with Aslan (the fact that Aslan is singing is likely a conceptual pun on the idea of the Divine "Word").