The word "Style" is used because of a law passed in the United States:

The Indian Arts and Crafts Act of 1990 was passed to protect Native American works mostly from Asian and other Companies creating works that were being passed off as authentic or made by Native Americans.

You can read on the regulations by clicking here:  Indian Arts and Crafts Act of 1990.

Keep in mind that there are many people who have some Native American Blood but are not recognized by the United States as Native American.  (In fact, there are a handful of tribes recognized by States but not Federal - the Indians that helped the Jamestown Colony have not been recognized federally - but hopefully they will this year).   Many Natives wanted to have nothing to do with the government (tired of lies and broken promises) and their ancestors fled into the hills or wilderness and stayed under the radar and are refered to as "Undocumented Indians".    The is act does not cover anyone that whether they are full blood, mixed blood, thin blood or whatever, they are not recognized.

Some Flutemakers (native and nonnatve) got in touch with that board and it has been agreed that the word "Style" added to the title is a legitimate way of identifing the product as a Native American Flute but the word "style" added indicates that the person is either non-native or might be a native that is not recognized by the BIA (Bureau of Indian Affairs)
 

I myself am not documented with any federally recognized tribe in the US.  I have some Cherokee and Lakota roots in my family and I fellowship with Natives around the Little Miami Valley and others on the Internet -- that description requires that I used the word "style".