Critics say if people were genuine about celebrating the victory of Mexican forces over the French at the Battle of Puebla, then they would not be dressing up in sombreros, donning fake mustaches and getting fucked up on tequila. Cinco De Mayo is cultural appropriation at its most obvious. It's taking things from Mexican culture, dressing it up as a "celebration," and using them in the most insulting and least respectful way possible.
There's nothing wrong with celebrating Cinco de Mayo, but the dominant ways that Americans celebrate it is horribly racist. Here's how Cristina Arreola at Bustle puts it:
Treat yourself to a plate of tacos. Listen to some Mexican music. Imbibe in a refreshing, frozen margarita. Go crazy with a round of bandera shots. But don't, under any circumstances, do any of these things while in brownface or while wearing a sombrero, serape, or fake mustache. Don't bust out your maracas. Don't dress up as a chili pepper. I promise you can have just as much fun in jeans and a t-shirt.
Others are pushing back, saying it's a bunch of politically correct nonsense that people can't celebrate Cinco de Mayo how they choose. It's a free country; tacos are good; sombreros are cool; and tequila is 💯.