Thanks for the video man, oh the memories.
Off the top of my head without going into much detail or trying things out i'd think there would be a few ways to do this.
Basically you need an image to be in the back and maybe other cut pieces in the front, and you can use shapes, colliders, empty bodies, invisible empty meshes, whatever you need to block the path or create the illusion of 3d.
One way I think should work to create the fixed image in the bg is to use a worldEnvironment and set the bg to texture.
So it's a different type of skybox. :D
But you can really create whatever illusion you want. You could render things into separate viewports, have a fixed image and transform the character in relation to the distance form the camera(screen), creating a fake perspective effect, anything.
Also, maybe you dont want the characters to change size into the distance then it gets simpler.
Whatever the case, what you want is a fixed bg with 3d characters in the front, moving as if there was a 3d environment around them.
Hope this gives you some ideas, you're gonna have to try things out and find your way. Good luck!