The basic answer is because of Cherokees having free instead of enslaved status and the rising trend in the 19th century to view Indians with some romanticism, even limited admiration. But yes, most white folks claiming native ancestry are misguided... Especially after removal, the tendency was to perceive Indians as the “noble savage” of a bygone era; no longer a tangible hated enemy but an mythic, tragic figure. The Cherokee in particular had been bitter enemies during the Anglo-Cherokee War and American Revolution, then gaining some respect (but not enough compassion or protection from the government) through their US service in the Creek War. Yet real Indians who remained in the Southeast had somewhat of an anomalous legal status (sort of in limbo in the perceived racial hierarchy below whites but above blacks)— being regarded as “free people of color”, most would have resented this classification, urging tribal “blood” and kinship ties over skin color. The terminology of blood, however, served to disfranchise and stigmatize people with native ancestry. Land ownership was sometimes possible but voting was pretty much out of the question. Furthermore, black slavery had fundamentally transformed Cherokee society, as a vector for increased acculturation and even emulation of the white planter class (see the example of James Vann, who owned a vast plantation and many slaves). The “one drop rule” applied to African ancestry was different than those with comparable native ancestry, allowing for more flexibility and social mobility (see the McGillivray clan of Scots-Creek ancestry).

Historian Christina Snyder has suggested that the persistent myth of whites thinking they have native ancestry may originate with people of native backgrounds being adopted into white families, often living in quasi-servitude and not actually “mixing” with the bloodline (She points to the example of Lyncoya, Andrew Jackson’s adopted Creek son). She points to a study where something like 40% of modern southerners claimed Indian ancestry while only 20% claimed Confederate ancestry. Thus, family stories handed down lost their concrete facts and became more vague or more embellished. At some point, it became somewhat fashionable to claim a “Cherokee princess” grandmother, perhaps to mitigate “white guilt” of colonization, or perhaps to suggest a male ancestor’s sexual conquest.

It’s worth noting, however, that there was more mixing and intermarriage between whites (especially traders) and Indians in the 19th century than most think, especially in western North Carolina where the Eastern Cherokee remained after Removal. Often this was white men taking Cherokee wives, but even in the 19th century some Cherokee men took white wives (Nimrod Jarrett Smith, for example, a sergeant in Confederate service and postwar chief of the Eastern Band). In the 19th century, western North Carolinians had a positive impression of the Eastern Cherokee, petitioning on their behalf, acknowledging their temperance, acculturation, Christianity and industriousness to prevent subsequent removal attempts. The vast majority of the Eastern Cherokee served North Carolina during the Civil War, further improving their local reputation.

Also, the Appalachian “Melungeon” people with fair complexion stressed a tri-racial ancestry to avoid being classified as “mulatto”. On the flip side, some people regarded as “free blacks” were actually people of mixed ancestry but descending from the Tuscarora/Lumbee and Chowan peoples in eastern North Carolina. They had to continuously fight for distinction as actual Indians rather than vaguely “free POC,” and though this afforded them few privileges, it was hoped they’d be more accepted or trusted to be left alone by white society.

This coming from a Southern whiter-than-sour-cream guy who studies Eastern Cherokee ethnohistory, who has never claimed native ancestry and hasn’t heard any such family stories. Yet took a recent DNA test which pointed to small percentage of SE Asian and Subsaharan African DNA, as well as some “unidentified”. There was far more “mixing”, especially in the colonial era, than most white folks realize (or want to admit).

For further reading:

“Mixed Blood” Indians: Racial Construction in the Early South by Theda Perdue

Slavery and the Transformation of Cherokee Society by Theda Perdue

The Eastern Band of Cherokee Indians by John Finger

“Andrew Jackson’s Indian Son” by Christina Snyder, in Native South by Greg O’Brien and Tim Alan Garrison, eds.

“From Indians to Colored People: The Problem of Racial Classification and the Persistence of the Chowans in North Carolina” by Warren Milteer

Edit: added an article and a sentence. Does someone have good sources regarding Melungeon identity?

I also should have mentioned that some Lumbee and others were erroneously referred to as “Cherokee” during the 19th century. Though all “tribes” were amalgams and coalescent societies to an extent, they were separate groups. So while a larger number of people may have native ancestry, not all of those who claim Cherokee “blood” are actually of Cherokee descent.

Also, to be clear, we’re talking mostly about the 19th century here. Some Cherokee and people of other tribes were enslaved in the 17th and 18th centuries— so their status as “free” wasn’t a given until the predominance of black slavery (late 18th- early 19th century).

Also, check out this Slate article