Humans to date
Humans have a distinct and recognisable face to allow other individuals in their lives to recognise them. Humans were jewellery, clothing, and various forms of skin markings to increase their recognisability. They will even adapt items meant to help surviving the environment such as boots, gloves, armour, glasses, bows, hammers, houses, etc... to be apart of this recognisable identity.
All of our social structures require a way to identify an individual so that their reputation can be tracked. This includes attributing praise, blame, privilege, and duty. Without the ability to associate a reputation to someone you do not know if that person is trustworthy or dangerous to you. The only personally safe attitude would be to treat everyone as deadly raving killers. You won't get a lot of society in these circumstances.
Losing your identity now days is deeply problematic. It makes it hard to open a bank account when the bank believes you already owe them an absolute fortune. It makes it hard to get a job, when the police believe you are a criminal. It may even threaten your life as others come to claim what your reputation owes. Regaining or even keeping your identity requires a lot of hard work to re-establish your reputation and prevent others from trying to take/tarnish it.
For the most part having a reputation is so important to attaining privileges, that it even makes sense to wear some blame, and deal with some duties. Like being dressed down for being late (to keep the job), or taking out the trash (its not pleasant but the alternatives aren't better in the long run).
On the other hand there are times when being associated with a reputation is itself going to be a problem. Like being able to eat 3 meals a day in a poor neighbourhood, or being the boss in a room filled with discontent employees.
So this race of shapshifters will actively fight to have independently recognisable identities. That may not be their face, it could be their voice, or some other hard to replicate thing. Most of them will actively dissuade others from mimicking them on this/these identifying point/s. Those that copy exactly would be considered reprehensible because of identity theft. This would draw social pressure for one of them to change. It may not change in the originals favour though.
Some shapeshifters will try to be unidentifiable because they are avoiding punishment, or enjoying a life of non-socially endorsed activities. They might do this by adopting a similar appearance to others, or by adopting several distinct identities based on location/situation, or by never settling on a single look.
Whatever isn't involved in their identity will be changed day to day with the seasons and whim. But this won't be a lot. Most choices of shape will be interpreted by many as joining a specific group. If that shapeshifter does not personally identify with the group, they will have to bear the social misunderstanding, or change to something else.
Within a group individuals will want to be different. I could imagine some shapeshifters choosing fairy wings, others would associate those wings with that groups reputation. Yet each shapeshifter with fairy wings would have something different. It might be their hair, their height, whatever it is any reasonable observer could tell them apart.
The only time the shapeshifters might literally clone themselves is in forming together to express a single demand of society or a single service to society. Militaries, protest groups, police, and judicial figures leap to mind. Here personal reputation is not important, but professional/group reputation is key. When the group succeeds all of the members succeed, when the group fails all of the members fail. The strict group identity will disappear though the moment an individual will benefit/fail independent of the group. The more the individual benefit/failure anticipated due to the group the less resemblance to the group.