It's not quite accurate to say that you're displaying normals. What's more accurate to say is, you're using the solid view preview, using matcaps aka sphere maps, using the matcap that looks like a normal map. That's not necessarily quite the same thing as displaying normals.
We can see that it's using that matcap, but that you have the "Color" on the overlays set to "Material", and you've specified a viewport color for your material, over there on the properties view. So what Blender is doing is giving you the output of this matcap multiplied with your material's viewport color. When you make a new mesh, it doesn't have that material, so that matcap will be multiplied with a different color, probably, white.
That may have seemed like splitting hairs when I was saying this wasn't technically a view of your normals, but maybe now its more clear why this wouldn't be a good way to, say, make a fast render of a normal pass. If you want something that actually is a view of your normals, you can switch to a material view and set the render pass to "Normal":