Editing Trans

Being Transgender and Transdynamic
In the version of Omegaverse I've settled on, it can be difficult to gender a baby at birth, unless you're using very modern methods like DNA testing. (I decided on this because even in our world gendering babies can be a challenge, and is based entirely on external genitalia. There aren't any other visible differences in babies! If omegaverse leads to more babies with ambiguous or dual genitals, gendering them becomes really hard really fast.) About 50% of babies aren't visibly male or female. After puberty, this number drops to about 10%, but it remains a lot higher than our world's 0.2-2%. 
So pups aren't really thought of as gendered. In our world, in the West, up through the Victorian era, kids of all genders often wore dresses until somewhere between 4-6. No reason to put a kid in trousers if you needed to change his diaper every 2 hours! In the omegaverse west, unisex child fashions extend up to 10-12 years of age. Gender-neutral pronouns are used. "Gender" starts with puberty.
Which means that trans(-gender) kids often have an easier time coming out as their correct gender. Even if they're part of the 50% of kids with a visible gender from birth, it was still only visible to people who were seeing them undressed. Everybody knows someone who was assigned a gender as a baby, and had a different puberty. Culturally, people aren't going to comment if you state you're a guy. Your family might comment, and puberty can still make things difficult, but it isn't a political issue in the way it is here.
Being trans-DYNAMIC is a different issue. (As is being gender-nonconforming.)
Puberty has the same effects in the omegaverse as it has here. Breasts, beards, body-hair, stinkiness, adam's apple, muscle-mass, height, etc. The effects are visible, and if you can match those effects, at least somewhat, people will classify you as that gender.
Dynamic, however, ISN'T visible. Male omegas tend to be shorter than other men (because a lot of their energy goes to building a uterus, rather than growth), but that is a only a VERY rough generalization. They have dugs (multiple sets of nipples down their torso), which are pretty flat to their skin and only obvious if they go shirtless. Otherwise, they look just like other men. Alpha women don't even have THOSE small differences. If you aren't looking for a bulge, there's no real way to tell.
Dynamic IS, however, extremely culturally important. Unlike with pups and gender, knowing the dynamic of a potential partner or rival can be vital. At some points in the past, different dynamic-gender combinations had distinct fashions. In the 20th century this became a lot less common in the west, though looser tops for male omegas (often with columns of pouches that small pups could be carried in to nurse) still show up. (Bras might not be necessary, but tight shirts CHAFE.)
In my world, there are still different titles and pronouns for various gender-dynamic combinations, and introducing yourself (or your companions) with the correct title allows people to properly categorize you. If you DON'T want to be 'properly' categorized ... that can be an issue. 'Lying' about your dynamic is seen very negatively - as a way of taking advantage of others. Stereotypes still exist, sexism and dynamism still exist, gender and dynamic-roles still exist.
Family structure and partnered relationships are a bit more flexible than in our world. Litters are large, and it helps to have something larger than a 'nuclear family' to support them. Non-child-producing relationships are useful - they often raise some of the pups of a pack, or financially and emotionally support those who are. Still, there are 'right' ways to have a relationship. And there's a HIERARCHY of relationships, with male alpha/female omega, and alpha/beta/omega, at the top. 'Pretending' to be in a more prestigious relationship type is deeply frowned upon.