According to Lewis Lathrop, longtime D&RG engineer from the 1880s on to retirement and quoted by his son in their work
Little Engines and Big Men the shape of the number plate was based on merit. In the early days of the railroad, locomotives were assigned to an engineer and his fireman and all routine maintenance work was performed on "their" locomotive by these men. If the engine worked, the men worked. If it was in for backshop repairs it could be a long spell between paychecks! Master mechanics appreciated when a locomotive was treated with respect and elbow grease, showing up for annual repair work in good condition with little for the shop to do save boiler and machinery repair. A crew that took good care of their engine received a brass rimmed round number plate. The men worked hard to earn them as this was a symbol of pride in the locomotive and their service. Engines that came into the shop with obvious neglect in appearance and mechanical condition (worn valves from lack of lubrication, tobacco juice stained cab interiors, etc.) were given a rectangular number plate. To lose the round plate was an embarrassment to the crew.
This practice worked well for the maintenance of the fleet of locomotives but it was discontinued around the turn of the 20th Century when the engines were pooled and no longer assigned to a specific crew. From then on, whatever number plate style was on the engine at that time usually remained until the end of service.