Well, one reason people think this is “true” is because they make better razor blades these days.
Blades used to be made of carbon steel and went dull very quickly. People learned that you needed to change your blade once a week or suffer the consequences.
Enter stainless steel (and later, other compounds). Blades lasted much longer. However, the razor blade manufacturers never bothered to tell anyone that (for obvious reasons). So people kept tossing their blades long before they got dull.
Enter the pyramid crowd. “Your blades can stay sharp a much longer time,” they said. And it was true. The modern, stainless steel blades could go much longer without sharpening – but that had nothing to do with any pyramids. People would try the pyramid and, for the first time, keep using the blade until it got dull. It lasted weeks. Miracle!
I have a tough beard. When I was shaving, a blade lasted me 3-4 weeks. No pyramids anywhere.
AFAIK, there has no been any scientifically valid experiment to determine whether there is any truth to the assertion, so it has to be considered false until its proponents can back up their claim.