Considering I have never seen an old trailer with the phrase “tool shed” painted on it, I can shed no light upon your question. I have lived in multiple states during my 40+ years, and have no idea what you are talking about.
In many places, you can’t have unregistered vehicles that are inoperable visible on your property. By labeling the trailer as a shed, he’s making a claim that it’s no longer a vehicle, but is now a structure, used for storage, which is likely allowed. As a prebuilt “structure,” it may be exempt from a building permit for it’s “installation,” and allowed on the property.
Thin, but thin often is all you need to satisfy a rule.