I can offer a brief answer relating specifically to Virginia.

There is a long tradition among the elite, old families of Virginia claiming they are descended from Pocahontas, who married the colonist John Rolfe (who famously developed tobacco as a cash crop, serving as the early economic basis for the Jamestown colony). These families are called the First Families of Virginia. These families took great pride in their claimed ancestry, though of course the veracity of these claims is doubtful. Even if these claims were true, tribes in the U.S. do not emphasize genetics or "blood" as an important factor in tribal membership.

There are a number of motives for why these families claimed this ancestry. Critically, they did not claim Native American heritage generally; they were specifically invoking ancestry from perhaps the single most famous Native American in white America's popular imagination. They are creating a connection to a foundational moment in Virginian (and American) history. For these families, claiming Pocahontas as an ancestor placed them as one of the very oldest clans in the country, and tied them to a pivotal event that solidified white presence in America. There would have been no shame in claiming ancestry from such an important figure so far in the distant past. Things would likely be different, though, if a scion of one of these families wanted to marry someone visibly Native American in the early 1900s.

In 1924, these families' claims of Native ancestry ran up against new laws that Virginia passed to preserve "racial purity." White supremacists, eugenicists, and others teamed up to pass the Racial Integrity Act in that year. This law was designed to prevent interracial marriage and reproduction by requiring citizens to register their race with the state, and defined a white person as someone who had "no trace whatsoever of any blood other than Caucasian"--a "one-drop" rule. This, the law's proponents argued, would prevent the white stock of Virginia from becoming "mongrelized" by "inferior" races.

However, this law posed a real problem for the elite families of Virginia, who were horrified by the prospect of no longer being seen as "white" by the state thanks to their claims of Native ancestry--no matter how small. Therefore, the one-drop rule was adopted with a critical exception: the "Pocahontas Exception." Whites with 1/16th or less of "Indian blood" would still be considered white. Thus, the elite families of Virginia could continue claiming Pocahontas an an ancestor, but continue deriving all the benefits of the legal status of whiteness.

In addition, at least in Virginia, there was a good deal of prejudice against people with Native American ancestry. A report entitled Mongrel Virginians: The Win Tribe (Win stood for White-Indian-Negro) was published in 1926 by the Eugenics Records Office, and contains incredibly derogatory descriptions of Monacan Indians, the real name of the group being studied. This deeply racist report sees these people as having a mix of White, Native, and Black ancestry. The researchers viewed this mixed ancestry as worse than being Black; they also were generally skeptical of the Monacans' claims to being Native American.

Source:

Elizabeth Catte, Pure America: Eugenics and the Making of Modern Virginia (2021).