Japanese is an oddball of a language (from an indo-european languages standpoint, don't know any others) in that second person is generally gender-neutral (no he/she), but first person is not, the existence of a plethora of gender-specific "I"s being the most striking example[1]. You could probably go for "watashi" though, as I understand it it is more or less gender-neutral[2]. Of course, anime and manga like to spice it up, with lots of characters having "odd" speech patterns, and combining that with characters scarcely gendering each other you get lots of characters of unclear/unspecified gender, which is pretty neat.
How widespread the notion of the nonbinary is I'm less sure about. Sure, characters of unclear gender are not rare, and gender roles have been played around with in popular media at least since Osamu Tezuka's Princess Knight, but Japan is also a very conservative country. My guess is that while the notion of gender neutrality and a tradition of playing with gender roles both exist, and so does of course a domestic LGBT+ community producing media with a higher concentration of such themes, the average person is still at best uninformed and at worst highly prejudiced.
Of course, one of the wonderful points about producing/imagining your own media is that you can put anything you want in it, disregarding previous tropes and standards.
[1] "Watashi", "uchi", "atashi", "boku", "ore", and "jibun" being the most common, probably. Using your first-name as a pronoun also occurs, mostly by kids. Then there's very formal pronouns like "watakushi" and "ware" and lots of lots of pronouns you'd only hear in anime, either because they are archaic ("sessha" used by samurai) or otherwise ridiculous ("ore-sama", adding a "revering" honorific to an already boisterous pronoun. Vegeta from Dragon Ball Z, uses this sometimes).
[2] Apropos, I've gathered there's an entire genre of so called "watashi novels" written entirely in first person with the narrator/MC's name not necessarily ever being disclosed. Pretty sure that e.g. the anime Tatami Galaxy and Humanity Has Declined were both based on such novels. Note that the formal has a male MC, the latter a female, both referred to using "watashi".