Transgender or Adhering to Societies Definition of the Opposite Gender.

[SuperNova]

As far as I'm aware gender dysphoria goes far beyond what you are describing.

While a more open and less rigid societal view of gender might certainly help, most trans people actively feel in the wrong body. They don't just like to dress and behave like the opposite sex they also often have major problems with their own body because it often behaves in ways their mind says it shouldn't.

Your profile says you're male, so just imagine for a second waking up tomorrow and suddenly just not feeling right. You feel unbalanced and emotional, that's because your testestorone to estrogen levels have just changed to that of a female (not being a dick about the 'emotional' part it's just what a major hormonal change does to you). Over the next few weeks your fat distribution starts to change, pooling around hips and ass. You start growing boobs. Just think about that seriously for a second (without the 'haha i can grope myself now' mindset).
Would that make you feel good? More than likely you'd be freaked out. You'd probably try to hide it as long as you could. Maybe work out like crazy or wear baggy clothing? This is what puberty for a lot of trans people is like from what I undestand.

Their whole self perception starts to feel wrong. Some actively hate their bodys.

It's been pretty well established that at least parts of gendered behaviour are there from birth and are pretty constistently found in one gender or the other. We are not sure yet, where exactly the nature vs. nurture ratio lies.
We know enough though to have at least some Ideas how gender dysphoria is caused (hormonal levels in the womb being at least a major factor) and to know that it is a real, likely irreversible, biological condition.

Less rigid social concepts of gender would probably help. Probably a lot. But they probably wouldn't eliviate most trans peoples need for surgery and artificial hormones eiher.