I feel like this pronouns thing started up in just the past few years. To me it seems more natural to say someone is a man/woman. If someone doesn't fit into one of those boxes then I think it makes sense to clarify the preferred pronouns.
That's kind of the point of using them though. The goal is to -stop- saying "you either fit into this gender binary, or you are 'other'".
It's kinda like normalizing the use of "partner" instead of "husband/wife" even for the straight married people; it helps normalize relationships that fall outside of monogamous, state recognized marriage, and doesn't force gender definitions (what if you're in a committed relationship, or married, to an NB person?) or force releasing sexuality just to indicate you're in a relationship (i.e., the person maybe doesn't want the fact they're gay to come up in this setting).
Providing your own pronouns, even if you're cisgendered, is a form of allyship; you're normalizing it, and helping create space for those whose pronouns don't match how they present to provide theirs.
But that said, what is super normalized is referring to others by their pronouns. "Becky? Oh she's..." etc. Providing pronouns in a place you don't need them avoids awkwardness, mislabeling, and normalizes it to where it's safe to ask for your preferred ones. I.e., "Hi, I'm Becky; my pronouns are she/her" makes it safe for those whose gender doesn't match what they present (or is ambiguous) to clarify how they wish to be identified, and is easy to roll with even for those who "aren't woke" or whathaveyou, whereas "Hi, I'm Becky; I'm a woman" would have no beneficial effect (it doesn't break the binary, since "non-binary" is still calling out the otherness from what is expected, and it also invites challenge by those against transness, i.e., exactly what we see GOP bills doing, demanding examination of a girl's privates and etc), and would likely cause you to be stared at like you're an alien for most situations.
It's one step further down the line from men and women being expected to behave in what are considered, respectively, masculine and feminine ways - the typical conservative viewpoint.
I find it quite bizarre that this is regarded as a progressive stance. It's like they've internalized the conservative view and come up with entirely new categories to work around it. When the sensible thing would have been to just reject the idea of gender stereotyping entirely.
So what a person is saying when they say non-binary can vary, and is also quite personal. Which is another reason to use the pronouns they provide (so you can respectfully refer to them using the categorization they provide), and to recognize that what is going on with their genitals, their chromosomes, etc, is not relevant to you.
Sex is a binary because that's how anisogamic sexual reproduction works: females embodying the reproductive role organised around large gamete production, and for males, small gametes. This still accounts for disorders of sexual development too: a male with underdeveloped testes is still male, and a female with non-functional ovaries is still female. In some extremely rare occurrences of chimerism, a person could reasonably be said to embody both.
But that's not what anyone means when they use the term "non-binary" as an identity - it's expressed by people who are unambiguously male or female, but who feel that they don't fit into the cultural gender stereotypes of masculinity or femininity.