At this point I don't even know what "identity" means. It certainly seems to have taken on a meaning beyond "The collection of factual demographic variables that I am" to some people and has morphed into something somewhere between a horoscope sign and a soul.
I don't "identify" as anything in the sense of making conscious decisions to be this or that. I am certain things that are objectively true. I am biologically male, left handed, a Type O+ bloodtype, about 5'9, 185 lbs on a good day. But these are things that are up for debate. You can't argue them. I don't "identify" as them as if it were a verb, I just AM those things.
Those things are part of my identity but I don't "identify" as them. It's not a verb, it's adjectives.
The trans side seems to be using "Identify as" as a way to take "I want to be" and codify it into some special kind of actually, literally being.
I think this has been posted in various forms before, but it's not a common bit that's been talked to death, so I'll toss it out here again
There are two commonly used meaning for "identity".
The first, and by far the most common, is the set of objective characteristics that other people use to disambiguate you from other people. These tend to be physical most of the time - height, weight, eye color, hair color, etc. These are the types of things that end up on your various identification documents, which you use to prove that you are you to officials. Sometimes, this sort of identity extends beyond the physical, in which case it's usually associated with either a career or a very strong reputation. Again, though, these are items that other people use to pick you out of a crowd.
The other, less prevalent type of "identity" is a psychological aspect. It's the set of characteristics that forms your view of yourself
for yourself. It's usually a combination of how you see yourself and how you want other people to see you, and it's usually based on personality and behavioral characteristics. For most well-adjusted people, how we think of ourselves is a fair reflection of how other people think about us - we are what we present to the world, we're honest in how we interact with others, and we're fairly objective about how we see ourselves. A lot of people have an element of psychological identity that is important to them, and it's very rarely something physical. It's usually associated with accomplishments, overcoming challenges, etc. Sometimes it's a little more nuanced. So for example, part of my psychological identity is associated with being an actuary, in particular with the type of thinking about uncertainties that comes with that. But another part of my psychological identity is associated with being a clear communicator. It's important to me that I feel that I'm understood, and to meet that need, I've put a lot of work into making sure that other people understand what I'm saying. I work with a lot of very complex information, with a high degree of uncertainty and varying degrees of materiality - being able to explain why my recommendation is a good recommendation, and support how I've come to that conclusion is vital to both my career and my sense of well-being.
I don't have the reference for this... but there's been some psychological research done that touches on these two types of identity. When we're asked to outline another person's identity, we tend to lean heavily on physical attributes (the first type of identity). When we're asked to outline our own identity, we tend to incorporate a lot more psychological and personality characteristics (the second type of identity). This is of course because we understand our own reasoning and thinking, and that's all opaque when we're talking about someone else.
There's a fair amount of the DSM-5 that is focused around Identity Disorders. In some people, the way they see themselves is in contradiction to how other people see them. These aren't perception disorders (like anorexia), where a person's perception of themselves differs from objective reality. These are based on the psychological characteristics that a person associates with themselves, but which are not in evidence by their observed behaviors and personalities. Identity Disorders often occur in tandem with various Behavioral or Personality Disorders, to greater or lesser degree.
The problem with the whole trans thing, to my mind at least, is that these are people who take their Psychological Identity and want to force other people to perceive them that way. How they view themselves is in contradiction to how other people view them... and they want other people to be required to adopt their own psychological identity as if it were reality. They think of themselves as having the psychological characteristics (which are often incredibly sexist in nature) of the opposite sex, regardless of whether they actually exhibit those characteristics or not. And they want other people to interpret their physical identity differently. So a male might psychologically see themselves as being delicate, feminine, graceful, sexy, sensual, submissive, emotional, etc. And because they believe that those psychological characteristic are an innate aspect of females (note that this is actually biological essentialism) they then wish for everyone else to see them as being female. In reality, most of the time, they aren't seen as being female by objective observers, and a whole lot of the time they're not perceived as having any of the behavioral traits they associate with females either. Many of them are quite aggressive and dominating, and are observably male. So at the end of the day, their "identity" becomes a force of coercion and control of those around them.