How do we define femininity and masculinity, and how to we apply such traits in a way that's not regarded as sexist?
Subjectively if at all, imo. The more precisely it is defined, the more exceptions that will occur.
The way I view it, masculinity and femininity are things that bring to us common mental archetypes. Masculinity is like, strength, assertiveness, hardness, coarseness, solidarity. Femininity is like, beauty, grace, fairness, curvature, gentleness, compassion. Most people get those types of images when hearing those words, masculine and feminine. They can refer to physical attributes (muscular angular build, vs softer more fluid build, for example) but in this thread and elsewhere it often references activities and behaviors (being a soldier vs liking to sew, for example).
But then the way I view that, is I take that rough spectrum of masculinity/femininity, and separate that from gender and sex entirely. So masculinity is a shared archetype about those types of traits we picture, but not linked in any way directly to men. Femininity is a shared archetype about those other traits we picture, but not linked in any way directly to women. The only indirect links would be that due to some combination of social and biological factors, statistically women end up further on the feminine side and men end up further on the masculine side as far as we use those terms.
A problem is that the etymology of the words "masculine" and "feminine" directly relate to the sexes. "Mas" means "male" and "femina" means "female". But in common use they refer to behaviors, ways of being, that can usually be adopted by either sex. So that's an ancient problem that persists with the words themselves- this idea of describing behaviors in terms of the sexes, rather than separating the classification of behaviors from the sexes, while still possibly observing that sure, the sexes do statistically concentrate in some behaviors more than others.
So I think that's antiquated, and that the spectrum of masculine and feminine still has some limited descriptive value for behavior and appearance, but not in a way that should necessarily reference the sexes directly. That way, if I'm a woman with 12 years of full-contact kickboxing and submission grappling experience, if people consider that to be "masculine", they don't have to directly mean, "like a man" or "unlike a woman" which would be sort of alienating to women (either by basically saying that a woman that does it is unwomanly, or reinforcing the notion that women shouldn't do these things). But instead simply share a thought about an archetype of behavior (hard, powerful, aggressive, whatever) that can apply to either men or women.
And even that's highly limited because a person could easily be, say, hard and powerful, but highly compassionate and gentle. The spectrum is nothing more than a quick description rather than anything deeply useful, if we must. Like if someone says, "he is effeminate", then I know probably not to picture in my mind a muscular cigar-smoking lumberjack of a man. That can simply refer to behavior and modes of appearance rather than having to be linked semantically and culturally to the sexes, though.
A related example would be height. Men tend to be taller than women, but we don't universally consider tallness to be related to men and shortness to be related to women. Like, a man is not suddenly "like a female" just because he's short. A tall female model is not suddenly "like a man" just because she's 5'11". He would simply be a short man, and she would be a tall woman. We could have had a historical situation where the etymology and semantics of the word(s) for height also directly referenced the sexes, but they do not. Similarly, words that describe archetypes of behavior, also don't/shouldn't link directly to the sexes in terms of semantics.
So overall my answer personally would be, why do we need to define feminine and masculine, and why do we need these broad umbrella words for traits? And if we think of them as an overly simplistic spectrum that brings to mind shared mental images of behaviors, then apart from etymology of the words as the currently are, why must we link those archetypes to the sexes?
Is striving for androgyny supposed to be the ideal?
I don't think so. I value the differences I observe between women and men as a whole; I just don't try to define it or institutionalize it. In fact it would be more accurate to say I value differences between something like masculinity and femininity, without necessarily linking masculinity to men and femininity to women. I'm glad for the diversity of life and character, but not glad about firmly linking certain traits to certain sexes with these excessively broad words.
I don't
need to define sewing as a masculine or feminine behavior. But that doesn't affect that I can appreciate diversity of appearance and behavior between people.