late edit: DISCLAIMER: The pictured map is not actually a representation of the territories before colonisation. It's a hypothetical map of what countries there might have been had the continent not been colonised, thus all the names and borders are fictional and have never existed.
For good actual maps, check out native-land.ca.
It's up to the tribe if she can claim membership or not. Many tribes require that you can prove direct descendancy from someone on the final rolls of the Dawes act. No amount of native DNA will gain you access if you can't prove direct descendancy. Tribes with casinos and casino money are especially guarded about letting people in, sometimes excluding groups who meet the criteria and have proof. There was a group of about 50 people who sued a tribe to be admitted when I was younger. They wanted in, because getting in meant you were financially set for life. They had all the proof needed to get in, but still lost the case. It's hard to win in court against groups that have billions of dollars, especially if those groups make the rules.
Right, Warren claimed Cherokee ancestry, not tribal membership. A story was passed down in her family and the DNA test results are in line with that story. She learned about the culture, visited the lands, and even published a cringey cookbook. By all accounts, her beliefs about her heritage were sincere and plausible, if embarrassing.