As someone who is latin american, I never understood what is the obsession with identity in the US. American hispanics I've talked to use it as something of a catch-all concept, but I've observed that many only have a meagre proficiency in Spanish and know nothing about the other Latin American countries that they pack with the bunch (apart from their own/their parents).
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Historically, a large part of it goes back to the Treaty of Hildalgo.
It obligated the US to enfranchise Mexican citizens living in the seceded lands of Alta California.
This immediately threw a wrench in antebellum politics which only allowed "white people" the vote. Eventually the US invented a racial category based on speaking Spanish...well actually two, one black, one non-black.
Fundamentally the identity of people with cultural affiliation to Spanish speaking cultures is complicated by the sanctioned narratives of US history which do not allow for the US to consider its Latin American heritage.
You could sleep in a hotel in Santa Fe and eat in a restaurant when the Pilgrims came ashore at Plymouth Rock. And when "fulfilled manifest destiny" reached the California coast, there were churches there where people went on Sunday...and chattel slavery had been banned for two hundred years.
This is a touchy subject in the US so I don't want to get too into it, its a bit off topic. The shortest and most neutral answer I can give is that minority political groups in the US use identity as a tool to push for political change.