in photoshop if you try to cut out at the middle of the edge, the edges look rigid. If you let anitaliasing in the selection that you are trying to extract, you get background color mixed. Which doesn't let it be on other colors.
So i know programitacially it's possible to extract transparency, like see immediate color of background, use that to calculate transparency of anti aliasing pixels and calculate and then use that.
But i can't think of a way doing it in photoshop this always leads to a dead end and lots of work. this is no problem is image is very high definition. Because even hard edge looks clean edge. Possible ways include cutting at hard edge but than fading edges. But this isn't a solution and not a possibility in all situations.
Or do it your way: extract this cat and bring it to red or white background.