He doesn't need to be a phoenix cause he already has Fawkes, so what's the point there?
Another thing I find out about phoenix is that they cannot be bought, sold or gifted away.
They choose the wizard them self and if the wizard agrees, a bond is forged between the two of them, kinda like till death does us apart.
And "they come to the protector of Light when Dark is eminent", so when Dumbledore was going to fight Grindelwald ( and conquered him ), that's when Fawkes attached himself to Dumbledore.