The enfacement illusion is a facial version of the rubber hand illusion, in which participants experience tactile stimulation of their own faces synchronously with the observation of the same stimulation applied to another's face. In previous studies, participants have reported experiencing an illusory embodiment of the other's face following synchronous compared to asynchronous stimulation. In a series of three experiments, we addressed the following three questions: (i) how does similarity between the self and the other, operationalized here as being of the same or different gender to the other, impact the experience of embodiment in the enfacement illusion; (ii) does the experience of embodiment result from alterations to the self-concept; and (iii) is susceptibility to the experience of embodiment associated with interoceptive processing, i.e. perception of the internal state of the body? Results indicate that embodiment is facilitated by the similarity between the self and the other and is mediated by the incorporation of the other into the self-concept, but sensitivity to one's own internal states does not impact upon embodiment within the enfacement illusion. This article is part of the theme issue 'Minds in movement: embodied cognition in the age of artificial intelligence'.