Aziraphale would not have admitted being an angel under normal circumstances. It was not something he would have admitted back home, in his world. Here, though? It was different here. People were more accepting, more willing to believe. Less likely to believe him insane.
"Yes. The demon is of my world -- my very best friend. The ghost is someone who is relatively new here. I met her when she arrived in town with her child. Not many people here are able to see her, seeing as she's a ghost. Crowley and myself can. We help her with her baby."
Crowley was actually quite good with children. And he had been more than willing to allow Annie and Eve to live with them -- he often allowed Aziraphale to do as he pleased.