I’ve personally experienced insurers having different “levels” of plans, and some employers pay for a higher level of cover than others. Once, I switched employers, my new employer perchance used the same health insurer as my old employer-but at a lower level of coverage.
I've seen and experienced this as well, but not to the point where calling the same doctor for an appointment would get me in earlier or later. More typically I have access to different doctors altogether, or need approval for certain doctors, or have different out-of-pocket costs.