The thing gender "actually is", when you pare everything else away, is a set of role scripts. Culture is made of rituals, and these rituals have labelled slots for people to fit into. The labels are genders (and also ages, social classes, and familial relationships, but that's less relevant here.)
Think of it this way: a wedding has a "bride" and a "groom." One way to think of it—the beach-head gay rights have established for us—is that either a man or a woman can be "a bride"; and either a man or a woman can be "a groom." Therefore, you could have a wedding with two "brides" who are both men, if you wanted.
Another way to think of it, though, is that being "a groom" (and all other such similar roles) is what it means to be "a man." In this model, a wedding is one of many rituals—a stage-play with a script—and one character in this stageplay is called "the bride", while another character is "the groom." You can't have two brides; that's not how the play of "The Wedding" is written. But anyone can play "the groom." And whoever plays "the groom" in the wedding, is also known as "a man"—at least for the purposes of the wedding script. If you happen to always play the set of roles in role-scripts that say that "a man" plays them, then you are "a man." Your gender is something you deduce from the set of roles you favor.
The work that goes into being transgender—the hormones, makeup, clothes, all that? Other than for the few people with true Body Dysmorphic Disorder (who would get reassignment surgery even if there were no other humans left on Earth to observe the results), the whole goal of the transitioning process is to get other people to "cast" you as the role you prefer to be identified with in these cultural role-scripts. A transman is someone who wants other people to intuitively place them as "the groom" at the wedding. That's the essential quality of it.
And, if you don't go for any of the available gender role-scripts that our culture defines? Then you don't have a (simple, obvious) gender, according to said culture. You would be genderqueer, nonbinary—whatever the word is today. Even if you're accidentally a completely central example of masculinity or femininity—if you aren't playing the role associated with that presentation, then it ain't your gender.