May 5th (by the way, that is not a typo – months of the year in Spanish are not capitalized) is NOT Mexican Independence Day, and is not widely celebrated in Mexico. It is not even a national holiday. What DOES it celebrate, you ask? Well…
It is the celebration of the Battle of Puebla, in which a heavily outnumbered Mexican army defeated the French army.
Mexico was in debt to Britain, Spain, and France after the Mexican American War (1846-1848) and the Reform War (1858-1861) had all but bankrupted the country. In 1861, President Benito Juárez declared a two-year suspension on the payment of foreign debts. While Britain and Spain negotiated deals with Mexico and withdrew their forces, France (Napoleon III, more specifically) decided to take advantage of the situation.
The French invaded the coast of Veracruz in late 1861, driving Mexican forces inland. On May 5, 1862, in Puebla a poorly equipped army of 4,000 Mexican soldiers defeated their French foes (an army of around 8,000).
You can read more about it on the History Channel’s website at: https://www.history.com/topics/holidays/cinco-de-mayo.
I’ll write about Mexican Independence Day in September, since it is on the 16th of that month.