Strange, I take the exact opposite reading of that quote: They do each have a gender identity regardless of form (or lack thereof), and can take physical form of their choosing. Yes they can change form (incl outward-presenting sex) -- that is, they have the metaphysical power to do so -- but AFAIK* there are no instances of an ainur choosing a physical form with a sex that is incongruent with their gender identity. I.E., they are always cisgender in the legendarium.

* I feel like there was one instance of someone taking a disguise of another differently-genedered character but I can't recall who it was or if I'm imagining this.

As for societal gender, it absolutely exists in Tolkien's world amongst most of the races/societies that we have exposure to.

Humans (and therefore hobbits) have gender roles generally consistent with that of European societies' "mainstream" that Tolkien was familiar with as an Englishman.
As an example for the Elves, male elves tend to be the ones that prepare food. As an "exception proves the rule" example, Galadriel partook in sufficiently male activities enough to be called Nerwen "Man-maiden", though I take this to be akin to describing a "tomboy" (but to the maximal degree, as Galadriel never does anything subtly).
There's definitely a gender division amongst Ents such that there are literally no female ents in the story except as a group which is explictly missing -- all Ents that are known to exist are by definition male/masculine in their society.
Dwarves are probably the closest to "no gender" in a Tolkien culture. But then again, every Dwarf which goes outside of their realms is said to be male so it seems like there is at least a gender role divide of "dwarf men go adventuring around the world; dwarf women stay in the city"
* As previously mentioned, there are Maiar such as Melian who married and gave birth to half-Maiar+Elven children, and also all of the Valar except 2 paired up and married each other in "heterogender" couples, maintaining the status quo of Tolkein's time amongst even the highest order of beings. It's hard to say how their gender identity expresses itself as societal gender roles since it is a such a small culture with only very specific examples given, but I think it is also generally true that male and female Valar fit vaguely into western European mythological tropes in a yin/yang sense: Yavanna (female) is "goddess" of trees and plants and her husband Aule (male) is "god" of mountains, crafting, etc.

* Edited to add some Ainur examples! We started this conversation explictly about Maiar/Valar/Ainur and I went off track to talk about the non-celestial races. Sorry about that.