Names have a massive significance for demons in Christianity - they hold power, and if you can name something you therefore have power over it. This is why priests ask for the demon's name during an exorcism - if the demon gives its name it is weakened. When the demons fell, God stripped them of their angelic names.
So when Crowley immediately tells Aziraphale, an angel, someone who should be his enemy, his name, it's not just him being friendly. It's him putting his faith in somebody he just met, saying "I trust you to know my name and not to use it against me." It shows how different he is from the other demons.
And when he changes his name to Crowley, he's rejecting the name Hell gave him. He's breaking away from who he was meant to be. He's the demon with an imagination and he's using it to forge his own path. And not only that, he immediately tells Aziraphale his new name. He's proving time and time again that he trusts his angel, that he would so willingly give him that kind of power. He's making himself vulnerable without thought.
And then again, during the church scene, when he hasn't spoken to Aziraphale for a century, he tells him his new name. He changes his name once more, moving even further away from the other demons. He takes a first name, something so fundimentally human, and still, even after such a long disagreement, he trusts Aziraphale with this vulnerability. This is the scene where Aziraphale falls in love with him. He is willing not only to enter a church for his angel and risk his life over and over, but to tell him time after time the one thing that could easily destroy him.
Crowley's love isn't shown just by saving Aziraphale from being discorporated. It's the smaller things, like telling him his name without ever doubting it. It's ineffable.
