The DNA of twins and triplets, although identical (for identical twins / triplets) at birth, isn't identical down the line, as mutations occur during growth.
I would say that even though they have the same DNA, what makes a person is based on the environment that they grow up in, and even though they may be brought up in the same environment, since they are separate entities, they will react in different ways to that environment. For instance, one of the twins might play with one type of toy all the time, because it manages to get that toy first. The other twin is then left with a different toy to play with. I'm not suggesting that toys have some big relation to how your personality develops, but I'm using it to illustrate that even in the "same" environment, you can be in a "different" environment. Kinda like a "sub-environment" I guess.
I would say that even though they have the same DNA, what makes a person is based on the environment that they grow up in, and even though they may be brought up in the same environment, since they are separate entities, they will react in different ways to that environment. For instance, one of the twins might play with one type of toy all the time, because it manages to get that toy first. The other twin is then left with a different toy to play with. I'm not suggesting that toys have some big relation to how your personality develops, but I'm using it to illustrate that even in the "same" environment, you can be in a "different" environment. Kinda like a "sub-environment" I guess.