Laying claim to culture is dumb, but everyone wants to feel special. One’s culture is less at risk from appropriation than from extinction as the world continues to trend towards an internet transmitted monoculture.
Laying claim to culture is dumb? That's kinda how things have been for the past thousands of years. A lot of these cultures have existed longer than many modern countries have. Yes, they do have risks from a cultural extinction, but watering a culture down or disconnecting it from it's roots doesn't save it from extinction. If anything, it advances that. Having people of that culture continue their cultural practices and protecting those practices, helps ensure that culture continues on.
Yes, there can be an internet monoculture, but everyone is part of some subculture. And the subculture is what is being protected. These cultures have existed for hundreds or thousands of years, and they carry historical importance in understanding history and the evolution of society.
Cultural appropriation is widely understood to be when someone in a position of power (most commonly a white person) utilizes something that is heavily related or socially seen as "part of" a minority community. For example, a white person wearing a Native American headdress. The issue that the minority community has with that is that they are routinely belittled or put down for participating in that part of their own community. For example, a white person wearing a Native American war bonnet to Coachella is appropriation, as war bonnets are culturally significant to Native Americans and you have to earn the ability to wear one. Meanwhile, many white Americans still put down Native American customs as "witchcraft".
When it comes to food, music, or clothing, you're pretty good there. It's primarily just avoiding things that are culturally or spiritually significant to a culture. I.E., don't wear a headdress if you're not Native American, don't act like you're the best rapper ever if you grew up on a farm in Kansas, or say you make the best gumbo in the world if you've never even been to New Orleans.