Before the mobs line up to lynch me, let me explain:
The problem lies not with the people but with the language. English, by its nature and heritage, is imprecise. We have too many words with the same or similar definitions and too many words that can encompass more than one definition. In this case, the problem lies with the word 'sex'.
'Sex' can be a noun and a verb - and therein lies the problem. If we can limit 'sex' to be a noun only, we get rid of such problematic words as 'transsexual' (where transgender probably originated), 'homosexual', 'asexual' and so on. Besides, we already have suitable replacement terms for the verb 'sex' (such as 'fuck, 'making love', etc.).
'Sex' - the noun - refers to a scientific, biological state defined by the 23rd chromosomal pair. A person's sex is either 'female' (having the 'XX' chromosomal pair) or 'male' (having the 'XY' chromosomal pair). The only ambiguity could arise from chromosomal mutation where the 23rd chromosomal pair is actually a triad ('XXX' or, more often, 'XXY') and such chromosomal mutation usually brings severe health issues. This even clears up hermaphroditism because most, if not all, hermaphrodites are still either 'XX' or 'XY', their genitalia being due to a genetic sequence.
Limiting 'sex' to being a noun only, it opens up a much more robust definition of 'gender' as a state of being (or sociological engagement). 'Gender', therefore, is a psychological construct (though 'construct' has troubling connotations, it's still proper here). Gender is the identity constructed by the brain - itself a genetic construct - through Nature as well as Nurture and has nothing to do with sex. If anything, since 'gender' is a function of genes influenced by experience rather than chromosomes, it is much more precise than 'sex' anyway. People are not 'asexual' (they can't be; everyone has a sex), they're agenderal ('a-' as a prefix meaning 'not' and thus agender/agenderal meaning someone who doesn't have a specific gender). People are not 'homosexual', they're homogenderal (which, honestly, doesn't make sense but is being used to build a point).
'Gender' is a construct but it is not, in most cases, actively constructed. It is no less real or immutable than sex, it's just built from a different, more precise DNA quanta and honed by life's experiences. This is not to say someone can't 'decide' to be a particular 'gender'. Since gender is much more fluid, it is inevitable humans will 'decide' to use a mask of a particular gender to facilitate their own agenda; potentially reprehensible, but it is going to happen. It happens with 'sex' it will (and likely does) happen with 'gender'.
More reprehensible is the fact some people will push a person who is not sure of their gender into being something they're not - the 'experiences' part of the Nature/Nurture equation. To identify 'gender' requires maturity. We should not require a child to identify as a 'gender' until they're mature enough to understand what they feel and think. That is not to say a child will not know their gender intrinsically, just that a child is confused enough about the world to add such pressure as having to define how they think or feel on their plate. Some will know from the moment they're born what their gender is - and that's fine - but let them gain some experience before they're forced to define it.
This brings us to 'transgender'. We are doing a monumental disservice by describing someone as 'transgender' because it implies they changed or have some control over their gender - their gender was 'a' (it is necessary to eliminate the stigma of calling a gender anything 'male or 'female' based) and now it is 'b'. If 'gender' is a psychological construct developed by (or imprinted onto) the brain genetically and guided by experience, then intimating they somehow changed their gender or 'crossed' their gender is denigrating their identity. The 'Kinsey scale' doesn't pertain to sex, it pertains to gender (and is much too narrow in scope to provide much relevance here).
Can gender change over time? It is certainly possible. In most cases where we use the term 'transgender', though, it hasn't. It just means we have not separated the noun 'sex' from 'gender' properly. 'Sex' (the noun) is physiological and immutably defined by the 23rd chromosomal pair. 'Gender' is a person's identity and is defined by their genes and honed by their experience.