I agree that the idea is a bit over the top, and as I said, I've never been asked for this.
But I think you might be missing the point.
Nobody is talking about a written reference - the idea is that you tell the escort you want to visit the name of another escort that you have been with. Then she 'phones her and checks that you are a nice guy. Your identity is established by your 'phone number - which many of the girls save. That's all.
They don't want a note from your employer