Plenty of manufacturers offered metal cans for grease before WW2, so it seems unlikely to me that it post-dated WW2. On the other hand, metal was a strategic material, so is it possible Standard Oil might have used wood to save metal for other wartime needs?
Most commercial grease containers show an SAE-rated weight for their grease by WW2, but this one does not, so again, that might be an indication that it dates from much earlier. The SAE (Society of Automotive Engineers) was formed in 1903 precisely to standardize things like viscosity ratings across the many manufacturers... so... could the lack of an SAE rating indicate that it predates the SAE? . I hope someone knows the history of this container and can solve this mystery.