the same generative fill background in different images

I'm not sure I am getting this.  The cat appears to be on its own layer with transparent background.  Is there a reason you can't carry on working that way keeping the layers?

If the original is a flat document, then copy layer and Remove Background.

Ctrl click the mask to select it, and expand the selection a couple of pixels.

Use Generitave Fill to fill in the cat hole in the background.

Keep everything layered going forward.

I feel sure you knew that, so what am I missing?

When you generate a background, hide the character layer by clicking the eyeball icon. Otherwise that character layer would get incorporated into the generated content. When the background is completely separate from the character, then you can toggle on/off the visibility of the character layers that you want.