Harris, 52, brushes off chatter about a future presidential run. Asked about potential presidential plans by CNN senior political commentator David Axelrod on his podcast, "The Axe Files," she said firmly: "I'm absolutely not thinking about that at all."

"There are a lot of very big issues that are on the table right now, and we have to be alert and present, like, right now," she said.

She wouldn't sit down for an interview for this story – one of the few ways she's actually following the script of a prominent freshman. But there's no question that her arrival in Washington at the same time as Trump may offer a helpful platform for her to gain national prominence.

No Democrat would have wanted a Trump presidency, but in many ways it's perfect for someone with the energy and talent of Kamala Harris," Burton said. "She is someone who has great instincts about how to get engaged in a fight.

At the same time, she is more reserved about her personal story than Obama was when he began positioning himself for higher office. While Obama wrote extensively about how the search for racial identity shaped his journey, Harris seems reluctant to engage on that topic, at least for now. When asked by Axelrod whether she gets sick of the anthropological questions, she laughed, admitting it's "uncomfortable" to talk about herself.

I was raised to do," Harris replied. "I was raised that you do, you don't talk about yourself, you just do. You don't talk about it after you've done it; you just do the next thing…. I would prefer to talk about what needs to get done, versus talk about myself.

For all the attention she generates, Harris is hardly on a glide path to higher office. She must prove whether she can harness her celebrity status into something more concrete. And plenty of her colleagues, including Sens. Elizabeth Warren of Massachusetts and Cory Booker of New Jersey, are making moves that leave the political classes buzzing with presidential speculation.

"It's all well and good, especially in this day and age, to have an active presence on Twitter and social media, but if you're really serious about things in the Senate, you've got to try to get something done or block something -- and for Senator Harris it remains to be seen whether she can do that," said Jim Manley, who worked in the Senate for 21 years and was a top aide to then-Senate Majority Leader Harry Reid.