I sometimes fantasized about having twins as a young woman and I've been told by many other women that they did too or would really want to have twins. It is fun, just plain fun - twins draw attention and the parents get into lots of conversations in public; they're cute; it is fun picking clothes, whether one chooses to dress them alike, coordinated, or different.
I've thought about their respective identities since they were born. I enjoy them being twins. They are really quite alike in physical look/structure, personality, and in their preferences. They pick the same colors, very similar clothing styles and mostly the same toys. They have always participated in the same activities/sports. Their conduct and grades in school are very much alike. They are consistently referred to as a pair - "the boys", "the twins", or their names strung together, ... and they are thought of as a unit by many people and/or in many situations. When they played football, if a coach couldn't tell which was which, they called them "Twin". It's understandable, especially with identical twins in uniform.
Yet, I want them to, and think they should, have separate identities. When they were babies and young toddlers, I dressed them alike, or alike but in different colors. As they got old enough to have and express a preference, I always used different colors. (I still bought two of each color so they wouldn't fight.) As I mentioned before, they like the same colors, but we could always settle on two. Now they help me pick out their clothes to buy and mostly pick out what to wear. I have the permanent right of veto.
Anyway, we strive to provide them different spaces and room to be an individual. They are in different school classes, different groups at day camp, and they are we encourage them to choose different activities and sports. We are currently moving our office out of a spare bedroom to make room for them to have rooms of their own.
A few other things that are interestingly alike - they cut their first tooth at the same time, they are both ADHD, they are within 5 lbs. in weight of each other at any given time.
In summary, I enjoy their "twiness" but guard against lumping them together. It is really hard when they are so alike and there are not other children in the family. Seems it's always about them, the two of them, and of course, the two ghosts IDAKNOW and NOTME.