The trivial answer is that its a political issue because she's a politician and a public figure, so its in several parties' interest to make a big deal about it.
Unfortunately, that also means its in a lot of people's interest to pretend things are much simpler than they are. I think its really important to start here with some basic facts.
The main one is that it is exceedingly ill-defined what it means to be Native American.* Part of this is that tribes historically did not operate on the European concept of "blood". The closest modern equivalent an internet user would likely understand would be gaming clans. If you wanted to join, and the existing members thought you could contribute and wouldn't be a cultural problem, then you were a member. This particularly appealed to clannish Scotts-Irish people. Sam Houston, while 100% Scotts-Irish was in fact a member of the Cherokee Nation with (for a while) a Cherokee wife. Chief John Ross, who led the Cherokee tribe on the Trail of Tears, was 7/8ths Scotts Irish, and had red hair and blue eyes.
So what a lot of people outside of Oklahoma may not realize is that here it is not at all unusual to see a Native American who looks completely "white". Particularly in Oklahoma, where Warren is from.
There also a disconnect with tribal rolls. They were created and families were put on them in the early 20th Century at the insistence of the Federal Government, but it was often fairly arbitrary how they were drawn up. Since there was money involved in being on them (government benefits, and sometimes tribal revenues), there has always been an incentive to get yourself on them, and to throw other people off of them.2 Excluded Native Americans take the tribes to court over this all the time, but there are many more who don't care that much about the benefits or don't have the money for lawyers and are willing to let it slide. So it is also not uncommon for an Oklahoma resident to be Native American (often from multiple tribes), but not be on any tribal roll.
So now let's look at the various parties involved.
She'd in fact very much like this to not be a political issue. However, since other people are making it one (and often in clumsy racist ways), she has to directly address it to try to get past it. Probably not going to work, but a person can try.
Her political opponents:
They may or may not know the nuances of Native American citizenship wrt Oklahoma that I went over above, but what they do know is that it doesn't take much work to make this look like a story of someone making a bald-faced-lie to the vast majority of people who have a stereotypical mental image of what a Native American is. This is why they don't necessarily discourage racist taunting of Warren. The more simple and stereotypical the discussion is, the better their case looks.
The Cherokee Nation:
This nation (tribe) is run by elected officials for the benefit of enrolled tribe members. They only get a certain pool of money from which to distribute benefits, so any argument that keeps their rolls from expanding to cover new families is an argument they are likely to be attracted to. For instance, the tribe owned slaves before the Civil War. The federal government has had to force them to add the descendants of those slaves to their rolls twice, because the first time the tribe kicked them right back off the rolls. Just last year this same tribe was arguing in court that accultured Cherokee freedmen descendents shouldn't be considered part of the tribe because they didn't share enough Cherokee "blood" (DNA), while this week they've argued someone with Cherokee DNA but no acculturalization shouldn't be considered Cherokee either. Its possible to claim that's a consistent position and a person ought to have a certain amount of both, but they only seem to bring up whichever argument helps them not add new people to their rolls. This is a very public case, so they could have a legitimate fear that someone would use this same kind of test to sue the tribe for membership. Again, since there are financial benefits to being Cherokee, perhaps keeping the rolls down is what they should be devoting their limited energies to curtailing.
1 - If I can be forgiven a plug here, my sister is an anthropologist working with the Osage Nation documenting how they are currently navigating trying to maintain a national identity in the modern world. The introduction of her book Colonial Entanglement has a really good primer on some of these issues of identity.
2 - With the Osage oil revenues for a while it was so lucrative that white people would trick illiterate Osages into adopting them, and then kill the Osage. The Osages eventually tried to clean up the mess by kicking all adoptees off the rolls (no mater what their culturization). I personally got caught up in that, which is how I can truthfully say that "I was an Osage for a couple of years in the 70's."