When you pick up the phone and someone asks for you, should you say "It is I" like you were an actor from a Shakespearean play or "It is me" like you had dropped middle school?
Every sane native speaker will say ‘It’s me’ rather than ‘It is I’. However, if someone asks for you by name on the phone, the normal response is ‘Speaking’.
Here’s what the authors of ‘The Cambridge Grammar of the English Language’ say about ‘It’s me’:
. . . if I’m outside on your doorstep and I call out It’s me, that isn’t an accidental slip on my part. It’s the normal Standard English way to confirm my identity to someone who knows me but can’t see. Calling it a mistake would be quite unwarranted.
Grammar rules must ultimately be based on facts about how people speak and write. If they don’t have that basis, they have no basis at all.
As a further statement of the current position, the following quotation from ‘The Cambridge Guide to English Usage’ is also relevant:
Traditional grammarians argued that It is I was the only correct form, whereas conversational usage has long endorsed It’s me. Research associated with the Longman Grammar (1999) shows that both appear freely in contemporary news and fiction, but in different constructions. It’s me is far more common when the pronoun is final (i.e. with nothing following it) , whereas the opposite holds when the construction has a following relative clause introduced by who, as in it’s I who suggested it.
On a very formal grammatical level, I in It is I is acting as a predicate because is is a copula, and so the nominative case, i.e. I, is correct.
However, in modern English usage, It is me is far more predominant, and only the most pedantic of persons would criticize you for using me in this sentence.
I think what you're looking for is "This is she" or "This is he". There are several ways to answer the phone, but this is the one closest to what you are trying to say. Neither "It is I" or "It is me" are used; "this" is used rather than "it".
As others have mentioned, "Speaking" or "This is [your name] speaking" (or even "This is [your name]" are also common and acceptable. But the closest to your original suggestion is "This is he/she", which is very common.