I'd be cool to see a special/different connection from birth! A lot of twins right after being born need to sleep next to each other as babies because they're used to being in the womb together. There's separation anxiety a lot of times if twins (or triplets and up) are kept apart. They also have the unique experience of growing up at the same time as their sibling so they really know each other the best out of most of the family. A lot of the personalities are the same I feel if the twins are close from the start, though I know depending on gender it can be different as well! I've noticed if it's two boys or two girls, alot of times there's a very close connection. I'm sure for boy girl twins it's similar too though I know it can vary from each pair of twins, sometimes boys are much different the their sister. I have heard of some twins not liking each other though which is hard to fathom haha but it's true.
Twins also develop their own language or way of communicating because they're so connected. Most times other family members won't pick up on it or if they do they don't understand it. At times there is telepathy but it could just be because the older they get and if their connection is strong it develops in it's own way.
A lot of times twins are each other's support systems as well as they grow up because they have someone who innately understands them on a core/soul level. They do say that some twins can feel if the other gets hurt as well.
They like to hang out and do a lot of the same activities together, but some twins develop their own individual likes, hobbies and interests the older they get. I have seen that there are twins out there that sometimes seclude themselves together to the point of antisocial behaviors (away from other people) and there are twins that keep the identity of being identical to a T until their in their old age.
Some twins marry other twins, or one twin is attracted to the same gender while another is not. There are also Siamese twins too! So that could be interesting as well haha.
I think other twins (and triplets, etc) can have different growing up experiences too if a parent favors one over the other. They also have similar dynamics as regular siblings where if a twin is born first and is older they might be protective of the younger one.