One-dollar coin bears the image of an eagle. The bald eagle is known to be a heraldic symbol of the USA. The eagle embodies majesty, power, victory, and according to ancient Greek tradition it is the direct emissary of the supreme god Zeus.
There is some justification for comparing two such distant objects as moon and coin, since the luminescence of the visible side of the moon is similar to the lustre radiated by the metal of the coin.
Analyses of samples of lunar soils*, brought to earth in 1970-1973, showed the crust of the ‘lunar seas’ to have high concentrations of heavy elements (iron, titanium, chromium and zirconium).
A well-known expression (see for example the stories of O. Henry) about ‘the moon shining like a polished coin’ is a metaphor, based on the observed similarity between such different ‘objects’ as coin and moon.
In addition, the ‘value’ of the moon changes, like that of the silver dollar, and its 100% worth corresponds, as it were, to the full moon.
Given the influence of the moon on people’s fortunes it may also be that the exchange rate depends on the phases of the moon.
*not sure one says ‘soil’ for the surface materials of the moon, but I can’t think of the specific word, if there is one. One might just say ‘surface’.