The problem is that all of your objects draw to the depth buffer and you use that to create your highlight effect. You will need some way of differentiating what pixels belong to highlightable objects in the shader, or a way to cull anything else out.
I recommend using stenciling. I won’t go into too much detail on how to use it, but I can help if you have questions. Basically, you’ll want to write to the stencil buffer when drawing objects that can highlight, then when you go to draw the highlights, only draw to the pixels that were stenciled.
If you’re new to stenciling, don’t be intimidated. Once you’ve got it figured out, it’s just a couple lines at the top of your subshader or pass. Unity - Manual: ShaderLab command: Stencil
I’m pretty sure the Rocket League setup isn’t working the way you’re thinking it is. It’s taking advantage of the fact it only cares about a single, spherical object, and the shader is calculating the intersection with a mathematical sphere with the position (and radius) passed to the shader. Otherwise there’s no way to do the intersection highlight in the areas the geometry isn’t visible behind the plane. Your test is a good example where the “glow” is only visible seemingly on the box’s surface. Valve did a similar trick in Portal 2 with multiple calculated spheres for when objects / the player pass through the fields.