A girl needs a room of her own.
— paraphrasing Virginia Woolf
It's important to control the amount of information people have about you, especially for creative tasks or in areas you expect you might want to change.
Defining yourself is risky, because then it becomes hard to change when everyone is relating to you as that. (Also for Keeping Your Identity Small reasons.)
Privacy helps avoid being defined as something, and it helps with managing how people define you when this can't be avoided.
A common idea is that other people judge you, assess you in their mind according to their standards, and there's not much you can do about how they see you.
In reality, people usually judge you according to the standards you give them.
People are complex — it's a lot of work to form accurate assessments. So when people tell us how to think of them, we generally believe them. (Often not consciously/intentionally, but what we think of ourselves is usually communicated in how we speak.)
All interactions require having a view of the other person and what they're about.
You need this to know what to expect from them, the role each of you play in each other's lives, and what they expect from you — how the interaction will go, what it will be based around, what's in the scope of the interaction and what's outside it.
So, it's a good thing we have some control over this!
We can have an idea of what we want out of the interaction and set the stage accordingly. If someone gets a false impression of us, we can intervene and correct it.
This has two difficulties.
One is that intervening when someone has a false impression takes effort. If someone has a lot of misconceptions about you, or if a lot of people have misconceptions, that can be arbitrarily difficult.
(This is related to "You can't please everyone": even if you could please anyone, it may be an intractably large problem to give every single person who knows of you a given impression. In practice, our impressions of people are never error-free.)
The other difficulty is: people change. Sometimes you no longer want the things your interaction used to be about. When people knew you as one kind of person, and you change, you have to manage their expectations — help them update how to understand you and your interaction with them.
If you're changing fast, this can be a huge amount of work. You may even change faster than you can communicate all the updates to everyone you have relationships with.
This comes up when you're newly developing ideas. Trying to figure something out, write a first draft, being in a transitional phase between two views, etc. Most people find it difficult to write if someone is looking over their shoulder. It's uncomfortable when you're just trying ideas and don't want them to 'stick' to you when they're just trials.
It also comes up when you're young. Children change very fast. This is the reason it's usually unwise for children's personal lives to be on the internet. Anything they post may end up defining them.
For example, if there's a photo of them playing with Star Wars, people might think of them as a Star Wars fan even if that was just a passing fancy that week. Which could also cause people to start relating to them according to stereotypes of Star Wars fans.
A similar thing happens when asking a 5 year old, "What do you want to be when you grow up?". This is a bad question. He might say "a doctor". Then at some point when he's 10, he'll have to explain why he doesn't want to be a doctor anymore, instead of just explain why he wants to be an engineer now. He has twice as much to explain.
And it's not just when he's 10 and wants to be an engineer. It's also next week when he wants to go to the bookshop, and he's directed to How Your Body Works instead of How Machines Work. Being defined in terms of "doctor" has now influenced his life in all sorts of ways. His answer to that question wasn't even his goal: it was just a guess, made in the context of his relationship with someone — he may not have even been thinking about it. (And rightly so.)
People still need to form assessments about you in order to interact with you. They need to know something. What should they be told? When should they be told things? Where do the lines get drawn?
I like the way you've started thinking about this.
You haven't addressed why we take on identities in the first place. Calling on a chesterton's fence - what makes you think people will want to change without knowing why identity exists in the first place?
At a very fundamental (but perhaps not useful) level, it's to communicate with people how to treat you and what to expect of you.
This might happen in some ways inside one mind (for example, if you have a self-image of being "a reliable person", that might help you think about how to be on time). But I suspect that usually happens when something is going wrong, because usually you can just think about the ideas themselves without needing to go meta.
What are the rival ideas about why we adopt identities? (I assume there's some background here I'm unfamiliar with.)
[quick link search]
Here's one of the standard posts on Overcoming Bias with a theory of identity, which I think just agrees with and fleshes out the first sentence of your comment (with more emphasis on the 'useful'). OB has a whole tag for 'identity' with a bunch of interesting nuggets.
Paul Graham's essay Keep Your Identity Small is a classic on this topic, and a quick googling finds these responses by LWers: Keep Your Identity Large, Keep Your Identity Fluid, Use Your Identity Carefully, Strategic Choice of Identity, Obvious Identity Fail. See Kaj Sotala's The Curse of Identity, and GothGirl's Three More Ways Identity Can be a Curse. If you're looking for clickbait, there's One Weird Trick to Manage Your Identity That Doctors Don't Want You To Know by the well-known self-promoter Gleb Tsipursky (okay I changed the title slightly). There was also some discussion of identity in the recent post Explicit and Implicit Communication.
If you're looking for discussions of the philosophical notions of identity and the implications of quantum mechanics (which you weren't) then that's a whole other list, but Three Dialogues on Identity is a possible place to start.
(Obviously you don't have to read them all to write posts getting your own thoughts in order. Though I would love to see someone write a thoughtful summary of the collective ideas and arguments in these posts with quotes - no need to add anything new, just distillation work. Perhaps we should also turn them into a sequence on identity.)
Edit: There's a large class of posts on identity and tribalism/politics by people like Bryan Caplan and SSC that I didn't include, largely because they generally didn't have 'identity' in the title and so didn't come up as quickly in search.
Identity is a way to signal your commitments.
As a toy model, imagine that people are playing iterated Prisoners' Dilemma game, and imagine that for whatever reason, people typically defect. Not always, but most of the time. You think that this is stupid, and that repeated cooperation would be better. But of course repeated cooperation requires the other person to cooperate too, and the other person's expectation is that you will most likely defect (because that's what an average person does). How will you convince them otherwise?
It is not enough to say right before the turn "hey, I am going to cooperate, and I hope that you will too, because this would establish a mutually beneficial long-term cooperation". Sure, it sounds convincing, but everyone is trying to say some convincing shit right before the turn, to convince the other person to cooperate; and them most of these people are going to defect. So most people will not be convinced.
You have to make a public statement in advance: "I am going to cooperate with everyone who has never defected against me. I am saying this publicly, so that you can keep records on my behavior, and verify with each other that I never broke my rule."
And if you succeeded to catch people's attention, and if they keep watching your behavior... and if after a while everyone sees that you really do follow your rule... now people have a selfish incentive to cooperate with you.
This "I am the kind of person who never defects first" is a simple form of identity. It is something you do in general (i.e. it is not an ad-hoc argument made up for a specific situation), and that is what makes it credible. Some people will argue that keeping your options open is always better for you, but as long as others see you as a person who keeps all options open, this is not going to work.
I prefer to reject the idea of personal identity. I am a collection of changing beliefs and desires, and so are you. It's a convenient shorthand to put more weight on the slowly-changing components, and there usually _is_ a continuity within a person (not everything changes at once, absent very unusual circumstances).
I do think that there are various persona I adopt in certain situations and with certain people - these are aspects of my current personality that I emphasize or slightly alter to better fit with my perception of other people's needs and comfort. But that's not identity, that's behavior.
Which leads to two additional tactic, available with different groups of people: