First, we need to consider why sexual dimorphism exists in the first place.
The first reason is because different sexes have bodies which are optimized for the different roles they have when it comes to caring for children. For example, in bees the sexually active females (queens) have much larger abdomens than the others. That's because they lay hundreds of eggs each day and don't do much else, so their bodies consist mostly of their reproductive organs. Spiders and some insects kill and eat the male after coitus to get the nutrition they need to make eggs, which requires that they are much stronger than the males.
Lack of gender roles wouldn't be effect but cause for evolving no sexual dimorphism, so we can assume that gender roles would be next to nonexistent.
A second reason for dimorphism is that it makes it much easier to find a partner. Most species (including humans) have little dimorphism as long as they are preadolescent and only develop sexual characteristics when they become sexually mature. So looking for people with the right sexual characteristics helps to find a partner who is a) of the right sex and b) sexually mature enough to procreate with them.
That means a species without dimorphism would need another way to separate partners from competitors.
An easy cop-out would be pheromones. But then we again have a different perception of males and females which appears to be against the spirit of this question.
One way would be a trial-and-error courtship behavior where everyone tries to get into the pants of a lot of different people until they find the correct equipment in them. This would be even more tedious and frustrating than our courtship behavior, so we can assume that emotional bonds won't form until finding out the sex of the partner.
Another would be to actually reintroduce gender roles. Sexually mature and available people will behave differently depending on their sex, which would allow to recognize them as such.
Would transsexuality exist? Good question. There is no consensus about why transsexuality exists in the first place. Also, there is no such thing as the "typical" transsexual. There are many different reasons why people feel that their biological sex doesn't fit them. Some trans people don't even know the reason themselves. But fact is, the less difference there is between sexes, both physically or culturally, the less there could be one could feel is wrong when they have the wrong biological sex. So when there still is transsexuality, we can assume that it would affect the daily lives of transsexual people a lot less.