I'm fact they don't share the future, they only share a starting point, after which they diverge immediately based on influences of the external world, like in this thought experiment the room the particular instance of the clone happens to wake up in.

The sequence of thoughts from the moment each clone realizes it woke up in room X, depends on the value of X, which potentially has lasting psychological effects that boost the divergence of the two clones. For example: imagine you've been told that cloning doesn't "split" you in two equally "genuine" clones, buy instead that the first one is the "original" whole the one in room 1 is the "clone". When you wake up and realize you are the clone, your thought process will probably diverge from the original.

Some people do. Dress them the same, even.

I would say treating them the same is an affirmation that they are different people. If they really were the same person, nothing would have to be done twice.