this is such a relevant question to me, yet i don't feel like i have a good answer. i like some parts of the other answers i have seen here.
i have a related question: is "identity" always somehow related to a group? can individual identity exist without group identity?
i see identity as simply how one sees oneself; or worse, how one defines oneself. (the distinction there being between dynamic and static). of course it also plays into how we see (or define) others. it consists of the entire realm of ever-changing characteristics, behaviors, limitations, desires, etc, that constitute who i am. it is difficult not to then use some label to (attempt to) serve as a shortcut for describing that identity - which is so nebulous, so dynamic, so absolutely non-static. yet, labels are essentially static. and unfortunately, the way identity is typically used in (so-called) radical circles, there is something very static about it.
which points to a distinction that i make. there is unique, individual identity, which is what i refer to above. every individual has some unique set/combination of attributes that makes up who they are in that moment. of course much of that comes as a result of looking at ourselves in relation to others. trying to come up with a meaningful label that can accurately describe all those attributes, much less continue to describe them as they change over time, seems pointless. so it makes sense that a name/nickname does as well as anything - it does not try to describe every attribute that makes up the individual, it merely labels the individual. the description comes from knowing that individual in any given context. and will almost always be something more complex than a simple label.
on the other hand, there is group identity. that is when a number of individuals share some (usually) single characteristic/attribute/desire/whatever. it is a common thread for the group, and it is the basis for whatever bond might exist between them all. unfortunately, in too many cases, their individual identity becomes subsumed by the group identity. everything unique about them becomes secondary to that one non-unique attribute. so the complex, dynamic, growing human being named joe becomes identified as "black".
when an individual defines themselves based on a group identity, that is where i tend to find dogma, ideology, reification, rigidity, and a general lack of individual freedom.
group identity is one way - maybe the primary way - we tend to see and define others, at least those we do not know individually. it is usually based on some visual physical marker. as dot has pointed out elsewhere, there are situations where that can be useful, no question (eg, someone in a cop uniform can realistically be assumed to have certain attributes, and being able to see that up front serves a very meaningful purpose for me). but far more often i find group-based identification to be detrimental to the kinds of relations i want in my life. stereotypes, predjudices, projections... these are what i usually see coming out of such group identity assessments. those physical clues serve to facilitate us putting others into rigid, pre-defined boxes, without actually knowing who those other individuals are. minimally useful to me.
at the risk of putting most of you to sleep, i'd like to use an analogy from computer (relational) database theory. [actually i am moving that to a comment below.]
"i both believe that it means something to be seen as being something (a race, a gender, etc), and i think it means less than/different from what identity politicians say it means. but i haven't found more clarity than that."
i don't disagree with that.
ultimately, who/what i am in any given moment is a conglomeration of innumerable influences. there are many group identities that may well factor into that; to elevate any one of those into a label that is supposed to accurately identify the complex individual that i am, seems (almost always) useless.
yes, i bear a cock, like half the humans on this planet. yet i despise being lumped into the group identity labeled "men". as i have said about both my wife and my long-term partner: she is twice the man i am, and i am more of a woman than she'll ever be.
it is the same reason why, back in my early teens, i refused to join the street gang that most of my friends were in. i hung out with them much of the time, did most everything they did; but i did not fly their colors or follow their rules. i rejected the idea of being subsumed by a group identity. yet, i did get the same hand-made tatoo they all had (albeit smaller than the rest), which (in retrospect) was clearly somewhat of a concession to that group identity.
wow, lots of words with little to say.