On Alto I use a BenDavis vocaltone which I've used it since the early 1980's. It's the only one I use on alto.
After listening to the track you mentioned the sound there is a happy convergence of, microphone, distance from microphone, ambience of the room, mood of player, reed happening to be in the zone, developing some small skill using recording techniques with audacity including picking and adding the right effect, getting some small proficiency in creating suitable backing tracks with BiaB and picking the right key that suits the sax and the voice.
I think if I recorded it again it would sound and feel different.