"Woke" is about shoehorning in extreme progressive political "traits". Whether that's in preaching the ideology, acting it out (turning white characters black, or adding extended lgbt scenes, etc), and so on. If you watched the barbie movie, what people mean by "woke" is mattel's behavior in the movie. In the movie, mattel's behavior is presented as "performative feminism" that's not genuine, but done for sales and PR/branding. The movie drags mattel hard for this and jabs at it several times. Mattel in the movie is "woke".
Sasha, at the beginning of the film, is 'woke'. She literally calls Barbie a fascist, and later "white savior". Her character is "woke" but realizes how harmful that is.
People saying Barbie is woke is baffling, because a large part of the movie is criticizing woke stuff.
Barbie literally started as a feminist icon. Literally go watch the movie it explains this.