The truth of the matter is not that everyone has a gender identity, nor that some people do and some don't, but rather that there is no such thing as gender identity in the sense required, namely of something that isn't sex but nevertheless somehow can 'match' someone's sex or not, in the latter case causing transness.
The existence of gender identity in this sense is a non-scientific myth; no-one has yet come up with any way of detecting its presence or absence. And more: until we have an explanation of how something so different from sex could conceivably match with sex, we are entitled to deny the sense of its postulated existence.
In short, there could be no such thing as gender identity in the sense Tatchell and others try to mean it.
So, no, Peter Tatchell, not everyone has a gender identity. No-one does. And you are talking abject nonsense. (Probably, let's be clear about this - and much of your public pronouncements tend to back this up - this is because you are just, well, a bit thick.)
[Of course people can believe what they like, however daft. Guardian angels, paranormal auras, devils, gender identities, fairies, immortal souls, gods, goblins, ... fill your boots. Just don't expect your beliefs generally to be taken seriously. And, most importantly, don't expect such beliefs to have any influence on public policy; not everyone agrees with you, you see.]