I have no time for Blair and I am not sure what his game is - other than a smug egocentrism.
I have not noticed Labour, or specifically Starmer , engaging in 'woke ' politics - this is a right wing press belief based on spurious to no evidence. This is disappointing from Blair, and specifically the Guardian to allow the word woke to be used in a dogwhistle DM way.
The establishment - especially the Labour one - need to exercise caution. So called woke politics on things like climate change, BLM and - yes- gender identity - are the core of young voters values. Young voters in their (not quite enough) droves aligned themselves with Labour under Corbyn. If they can be persuaded to vote, they are powerful.
The Tory government is engaged in a (terrifying to me) war against any liberal attitudes at the moment and this seems to be seeping through, including te belief that widening university participation at elite universities is somehow unfair and woke, for example and that the privileged elite are terribly disadvantaged. They don't believe anything - they just say what they think will run well with any core demographic who are a bit pissed off.
If woke is believing in equality, for all and ensuring structures in society work to the benefit of all then colour me woke and bury me in a woke coffin.
As a side note, why are lifelong Conservative voters never called 'hardcore'?
We have discussed on here whether the particular aspect of sex and identity is 'important' before in the sense of whether it matters as a top issue to most voters. I think that conversation has been done but various surveys of voter attitudes do suggest it is not a top priority. This does not mean it isn't important to individuals but MN magnifies its importance as an election . voting issue. This may, in part, be what Blair means - but I don't think labour are spending much time on being 'woke' n gender politics, or anything similar.