Some of the legitimate uses of "identity" are worth considering in detail, because they are often, by a sort of pun, allowed to lend spurious plausibility to the bad ones. It is therefore important to point out that they are not actually up to that job.
2.1 The logical sense
Identity in logic is the relation that, by definition, each entity bears to itself and to nothing else. It is needed in ordinary language & in most constructed languages because they contain synonyms (different names of the same entity) and descriptions (noun phrases that characterize an entity). In ordinary English that relation is usually represented by some form of the verb "to be" (which, however, has other uses). To say that George Washington was the first president of the United States is to say that the proper noun "George Washington" and the singular description "the first president of the United States" designate the same person.
A quirk of usage is worth noting. In most mathematical talk, the sign "=" and the corresponding word "equals" are used in the sense of identity. To say that 2+2=4 is to say that 2+2 is 4: that "2+2" is a description of the number called "4". Equality does not mean that in ordinary English --- or in plane geometry as traditionally taught. There, to say that A and B are equal is to say that they are equal in some property (length, say); it is that property, not the entities, that is identical with respect to the two of them. "All men are created equal" does not mean that there is only one man; it means that all men have the same political rights. So also in geometry from Euclid down to my childhood: If, approaching the Bridge of Asses, I wrote that AB=AC, I would not mean that AB and AC were the same line segment; I would mean that they were equal in length (had the same length). A modern mathematician would disapprove, and would insist on my writing something like m(AB)=m(AC); I suspect that by now that scruple has found its way even into elementary textbooks.
It should be clear that identity in this sense is merely a linguistic convenience and has no bearing on psychology, sociology, or ethics. In particular it is not a relation between a thing and its parts. I am (near enough) a connected warm region of space-time bounded (near enough) by a birth, a skin, and a death. I am, by definition, identical to myself. My first year, my left foot, my nose, my self-esteem, and my fiftieth year are each identical to themselves; none of them is identical to any other of them, or to me.
2.2 The police sense
This sense is more complicated & interesting. I call it the police sense because it is what the police mean when they say they have established the victim's identity, or suspect a case of mistaken identity, or have provided the witness with a new identity, or are investigating a complaint of identity theft. It means a set of observable attributes that are held together by the knowledge that they belong to the same person. My name, address, telephone number, Social Security number, bank accounts, birth date, citizenship, and eye color, at a given time, are components of my identity in that sense. Given two or three of those, you may (perhaps with the help of a warrant) find out all the rest, and perhaps even find me.
There are probably a few hermits who have no identity in that sense. Most people have exactly one. A few have two or more, with the help of fraud or police protection, but that is hard to manage.
The components of a police identity are usually not of great importance one by one. What makes them (socially) important is their connection with each other. That allows me to ask "Who are you?" and mean "Tell me enough about yourself that I can find you again".