That's clearly quite wrong. Gender has a meaning that even in the most hysterically PC definition can only be defined by sex. Otherwise what does it even mean at all?
"Gender" is the social constructs that we typically associate with sex. And those can change, society to society.
Men in Greece wore white pleated skirts up until this last century (and even into the World Wars in formal military dress), but that particular aspect of their culture is gendered female to modern Western nations. The gendered performance of "male" is thus drastically different in both cultures, separated by time and distance.
Essentially, gender is the collection of expectations, attitudes, aptitudes, tastes, preferences, and affectations expected by society, it is the performance arising from our culture's views of gender identity. Sex is physical.
To put it another way, gender is behavior, sex is biology.
Most mainstream psychologists make a pretty clear distinction there, and I'm kind of surprised you haven't run across this difference, to be honest.
Many people who imagine that their gender conflicts with their sex realise this only too well, otherwise they wouldn't undergo a surgeon's knife to emulate a different sex (not gender).
...but not all do. Some are okay being a woman with a penis. Humanity, even that segment of humanity existing in a niche you and I may not have first-hand experience of, exhibits a remarkable amount of diversity.
No to derail things but where does the line get drawn? Is there a line? I mean, what if a 52 year old man identifies and lives as a 6 year old girl? Transgender AND Transage? And what about the Otherkin; those that identify as animals? Do we just accept every manner of self-identity as if it's ok? As if there is not something mentally wrong with people?
Ilness is defined by its consequences. A man who identifies as a six-year-old girl will likely find his life and future opportunities severely curtailed. So in that case, yes, at least in the modern day, that's unhealthy because of the consequences such a perspective can have. But "abnormal psychology" is not inherently bad as a result of its abnormality. It is a problem when it carries with it consequences that the person with the abnormal psychology would like to avoid.
The otherkin are adorable, and I don't begrudge them their fantasies. It's up to them whether or not they want to seek "help" for that, based on how much those beliefs about themselves are internalized and blocking their progress.
But I think with those two examples, the transage and transspecies business, there's an important difference with transgenderism. Our age changes throughout our life. It's kind of unavoidable. And our age is pretty easily objectively defined. It may start in a different place based on different definitions (you start at conception, I'm willing to bet; I start at evacuation from the womb), but unless you're orbiting near a black hole, your age more-or-less progresses in a way defined by your position in space-time. Liekwise, otherkin, when they actually do believe what they say (I've met some who were less-than-sincere personally), are believing something that is likely the result of suggestion or motivated reasoning, rather than something structurally in the brain.
But feeling like or believing you are a different sex, wanting to present as a different gender, has an explanatory mechanism. RD mentioned it earlier when she talked about situations where people develop a particular set of genitals as a result of in-utero contamination when they would otherwise have developed in some other way. The brain is a pretty delicate and anything that skews hormone levels, especially in early developmental stages, can have pretty weird consequences, consequences that we don't wholly understand.
We are too neutered as a society now to call out anyone on their nonsense and that leaves the marginalized and outcast to come up with new "identities" to show others they really are that special little snowflake, unlike all the rest.
Or they genuinely do feel out-of-place in their bodies, with the gender performances they have. Surely you don't think Olympic medalist reality TV stars need some external element to help them feel more special, for instance?