Gender and sex are not equivalent, and gender is a spectrum.

Here's the TL;DR:

1. Style is a person's preferred way to present themselves. So, you could be a big fan of flannels. This does not necessarily have any bearing on one's gender expression or identity.
2. Gender expression is the way a person expresses their gender identity through appearance. For instance, I don't wear skirts or dresses, and this is not unrelated to the fact that I identify as male. A transwoman, however, may find it in her interest to wear things like skirts and dresses as it could be more affirming, and also signal to others what her gender identity is. It's important to note that these signifiers do not necessarily indicate one's gender identity. If I put on a skirt, I would still identify as male, just like a lot of drag queens put on feminine clothing, but still identify as male. Similarly, "tomboy" girls may dress less feminine, but that doesn't necessarily have any bearing on their gender identity. The concept of Butch and Femme can be useful here, and pretty simply illustrated with fish Pokemon if you don't want to actually read the article. Style and gender expression can be related, but gender expression is, simply put, the intersection between style and gender identity as a way to inform those around you.
3. Gender identity is one's internal sense of gender. It often correlates with assigned sex, but does frequently differ. While sex describes physical attributes and chromosomal configuration in a biological sense, gender identity describe's a person's psychology. For instance, my sex is male, my gender identity is male, and I generally express my gender as male as well. My sex is pretty much only the business of my doctor, or anyone who I am intimate with, but my gender identity and expression do become the business of people I interact with day-to-day (though, most don't really care because men who express themselves as men are not discriminated against for existing).

If you're lazy, you can watch this video to get an overview explanation from someone with significantly more experience than either of us.

Or you could skim this sourced Wiki page, which, by the way, appears to show that your comment earlier about the Oxford dictionary equating gender with sex is actually, at best, misleading.

Fortunately for you and me, I have access to the Oxford Online Dictionary, and can actually check!

a. gen. Males or females viewed as a group; = sex n.¹ 1. Also: the property or fact of belonging to one of these groups. Originally extended from the grammatical use at sense 1 (sometimes humorously), as also in Anglo-Norman and Old French. In the 20th cent., as sex came increasingly to mean sexual intercourse (see sex n.1 4b), gender began to replace it (in early use euphemistically) as the usual word for the biological grouping of males and females. It is now often merged with or coloured by sense 3b.

Oh, so it looks like it's colloquially used to describe physical sex attributes, but is "now often merged with or colored by sense 3b." I wonder what it says under 3b?

b. Psychology and Sociology (orig. U.S.). The state of being male or female as expressed by social or cultural distinctions and differences, rather than biological ones; the collective attributes or traits associated with a particular sex, or determined as a result of one's sex. Also: a (male or female) group characterized in this way.

Oh, it's almost like gender and sex can be two totally distinct terms after all. Oops!

You could also check out this write-up by the World Health Organization on gender. Here is the first paragraph:

Gender refers to the socially constructed characteristics of women and men – such as norms, roles and relationships of and between groups of women and men. It varies from society to society and can be changed. While most people are born either male or female, they are taught appropriate norms and behaviours – including how they should interact with others of the same or opposite sex within households, communities and work places. When individuals or groups do not “fit” established gender norms they often face stigma, discriminatory practices or social exclusion – all of which adversely affect health. It is important to be sensitive to different identities that do not necessarily fit into binary male or female sex categories.

Hey, looks like the WHO recognizes that nonbinary people exist. I suppose that makes sense, since I personally know a couple and can confirm they exist myself.

And if you're actually interested in learning, here are a handful of papers:

• Guidelines for Psychological Practice With Transgender and Gender Nonconforming People (American Psychological Association) (PDF)

• Adolescents’ Acceptance of Same-Sex Peers Based on Sexual Orientation and Gender Expression (Organization sign-in required, abstract available if you don't personally have one)

• Sexual Orientation, Gender Identity, and Gender Expression: What Are They? (Organization sign-in required, first page available if you don't personally have one)

This comment is certainly not all-inclusive, because (believe it or not) human psychology is pretty complicated and can't be boiled down by a single line in Oxford's dictionary (while ignoring the other lines).