I'm note really sure what a good first approximation would be, but maybe I can break down the elements of it a bit, and we can see whether there is some other experience that mimics it.
* Spending several years, uninterrupted and undistracted, thinking deeply about a mathematical topic.
* Being advised and supported, but not led or coddled, by an experienced researcher.
* Being surrounded by clever people, who do research full-time.
* Working in an environment where not knowing the answer. immediately, or even quickly, is not just accepted by expected.
* Being "left in the dark", with the understanding that the process of being in that position and finding your way out is the very thing that you are trying to learn.
* Having rigorous expectations for the "final result" (to the extent that one can be found).
I can't think of a great approximation to that, but perhaps someone else can chime in.
EDIT0: Formatting
EDIT1: Also worth noting that by far the smartest person I know - a deep thinker, also nominally a data scientist, but foremost a truly original and creative mathematical thinker - didn't complete his PhD and left with a Masters. Intellectually, this guy makes me look like a geriatric Corgi on opium, and he didn't need a PhD to make him that way.