Firstly, my understanding here comes from the various people of colour I've spoken to about identity and politics- I am white and it is not, personally, my area.
It seems that some people, with Kamala Harris' heritage, would identify as black, Indian, mixed-race, or a combination of those terms, mostly depending on the community they were raised in and the demographic they're speaking to. In that sense, it's code-switching. Combined with cultural history, like the aforementioned "one drop rule" and the politicisation of black lives in the U.S.A., I can imagine that 'black' feels like the most relevant term to explain her experiences and the prejudices she is up against. As a PP mentioned, 'white' is often still viewed as the 'default' race in many Western communities, and as such people who are mixed-race white and black are far more likely to identify actively with their black identity than their white background. Kamala Harris does seem to throw a lot of people off in that sense, coming from two minority backgrounds and forcing people to eschew that assumed 'default.' Being Indian, which I imagine Harris considers equally important on a personal level, simply doesn't hold the same cultural connotations and context in the U.S.A. as being black does. That's not to say whether it should or should not, but at the moment it isn't as politically and socially relevant to her career and campaign.
There is also a level of negative connotation people feel towards being called "half" one race or the other, which is understandable. If you have grown up both black and Indian, and have been perceived by others as both black and Indian, have been involved in both black and Indian communities for your entire life, it seems strange for others to insist on the 'half and half' terminology. Obviously there is also a long history in Western societies of attempting to divide and categorise minority ethnic communities into what is considered easiest for white people to understand, which must be uncomfortable to feel reflected in current discussions. Technically, genetically, Harris is half black and half Indian. Culturally, and societally, she is black, and she is Indian.
All of that aside, I do think the wider media have a responsibility not to actively ignore or erase her Indian heritage and identity when she mentions it. I think the onus to remember, respect and not make assumptions about people's racial identity is mostly that of white people, myself included.