I am absolutely used to not mentioning my identity first if I am making the call. That is simply out of sheer politeness.
I was taught that the caller always identified himself immediately.
That's definitely the opposite order of the culture I was raised with. (Although, admittedly, I think I vaguely recall having heard heard that my grandparent's generation may have had that expectation.) The expectation I'd have is:
that the receiver finds out who the caller is trying to reach
- unless the receiver is even more polite by providing the receiver's name.
Then, the receiver tries to get the desired recipient on the phone, if possible.
If that's not possible, the receiver says so. If the receiver is willing to deliver a message, then the receiver can offer to take a message.
Once the desired recipient is on the phone and has identified themselves, then the caller identifies themselves.
Then, if the expected topic of conversation isn't assumed, the caller identifies why the call is being made.
This is most respectful of time, including professional scenarios (business calls) and less professional ones (when one child calls for another and reaches a sibling in the middle). The caller has already taken the time to call, wait for the ringing, and speaking to whomever answered the phone. By the time the desired recipient is on the phone, the "least that person can do" (to achieve minimal politeness) is to self-identify so the caller knows that the caller has completed the third phase of the phone call (after dialing, and ringing, the right people are now on both ends of the call).
When I answer a phone that is shared, I find it less polite if the calling person identifies themselves to me when I don't have any use for the information. If John is nearby, then why do I care that your name is Peter? I just want to know that you're trying to reach John, so I can tell John that there is a phone call. Giving me a second name, which I may not need, doesn't help me any. If anything, it might create additional opportunity for me to mix up the names. Why burden me with that? I'm doing you a favor of responding (so you know someone is at the destination location) and presumably will be polite enough to tell John that there's a phone call. So I'm doing you (as the caller) a favor. The least you can do is minimize my challenge in doing so.
If I decide that John isn't in, and I offer to take a message, then maybe I will be fortunate enough that you'll say "No thanks. [I'll call back later.]" Great. Then I didn't even need to take a message. I certainly didn't need your name. Thank you for not sharing it.
On the flip side, if I am making a call, my desired intention is to reach someone (which I made clear by causing the phone to ring). By asking for a person, I am clarifying who I am trying to reach. Frankly, who I am really doesn't matter until after I get that person on the phone. I really just need that person to have one piece of information: Who I am trying to reach. Giving two names (who I want, and also who I am) just takes additional time from a person who has already started giving me time by answering the phone. I'm not likely to rudely impose with my name, until asked. (And, if I'm asked before the person does the decent thing of letting me know if they can get that person on the line, I made be mildly annoyed by the slight impoliteness of the call screening.)
If someone starts asking me questions, like my identity, before giving any indication on whether they are going to cooperate with the common courtesy of getting the right person on the phone, then I am a bit surprised at the breach of what I believe to be the common courtesy (which I just identified). At which point, I internally sigh and start judging whether I am getting anywhere towards reaching my goal (of being able to tell the information to the desired recipient), or whether this person is going to waste too much of my time. If the efforts demanded of me exceeds the importance of communicating now, I might just stop playing stupid games, and try contacting the person later, possibly in person.
To this day, I answer the majority of my phone calls by doing the polite thing, which is to provide my name to the person calling in my initial greet. ("Hello, this is X.") The only times I tend not to do that is when caller ID indicates this is a person who will recognize my voice with a simple "Hello", or when I am answering a phone number with which I am likely to receive calls by multiple people who may know me by different names (since, to avoid duplicate name collisions, I now primarily go by my middle name, but many people know me from before I tended to do that).