Your small coin, dug in Mexico, saying "Emperador Maximiliano" on it, also shows a droop-winged eagle with the (French) crown above. It is from the 6-year reign of Emperor Maximilian 1, the only monarch of the 2nd Mexican Empire. In 1861, due to massive unpaid financial debts to several European governments, troops from France, Spain, and Britain deposed the Mexican Republic's government, with France taking the opportunity to create the 2nd Mexican Empire. France installed Archduke Maximilian (of Austria) as the emperor. He reigned from 1861 into 1867, when he was defeated by Mexican Republic forces (under Benito Juarez) after France withdrew its troops and support.
Sidenote:
Maximilian welcomed many thousands of Confederate refugees fleeing the yankee Occupation in 1865. Most of them returned to the US after Maximilian fell.
For more info:
https://www.historytoday.com/archive/emperor-maximilian-arrives-mexico-city