While some objects were designed to represent the thirteen original states, such as the American flag and the dollar bill, glass paned doors like the one above are probably not another example. Some pieces of furniture made in England in the Chinese style do have glass-paned doors with thirteen panes, but they certainly weren’t celebrating America’s independence. The furniture design book of Thomas Chippendale, which was published in England 23 years before the American Revolution, shows case pieces with doors that have thirteen panes of glass, like this one:
It’s a nice story but one totally unsupported by fact.
Sounds like folx who designed such things liked the odd-number aesthetic – usually a design with an odd number of elements is more pleasing than a design with an even number.