>I don't know why America thinks gun owners universally support the NRA

The only organization I've ever heard gun owners talk about when also talking about the second amendment is the NRA, and the NRA is the only organization known by the general public that claims to speak on behalf of gun owners.

Americans believe it because the NRA has been very successful at branding itself as the de facto voice of Second Amendment politics, and gun owners haven't put forth any effective opposition to that.