A relation that fails transitivity cannot constitute numerical identity, which is by definition transitive, so psychological continuity must be tracking something other than strict identity.

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Strict identity(in metaphysics and logic): Perfect, absolute sameness in every way—if two things have strict identity, there is no meaningful difference between them whatsoever.
Tracking(as used in epistemology and philosophy of mind): In philosophy, this means reliably responding to or being guided by something—like how a compass 'tracks' magnetic north.
Transitivity (or transitive)(as used in logic and identity): A property where if A relates to B and B relates to C, then A must relate to C. For example, if John is taller than Mary and Mary is taller than Sam, then John must be taller than Sam.
numerical identity(Distinguished from qualitative similarity when discussing whether a tailed cat and a tailless cat are the same individual.): The relation an entity bears to itself and nothing else; being one and the same individual entity rather than merely qualitatively similar.

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