From the OP:
“Without the consciousness of sameness, there would be no consciousness of transition.”
There is the consciousness of a correspondence - a continuity - between one experience and the next.
The perception of an unbroken continuity comes about through remembering some of the content of thoughts that have ceased.
If the memory fails the awareness of the relationship between what is happening now and what happened previously breaks down.
There is no meta-awareness of passed and present states that is conscious of both. There is a sequence of experiences that includes a remembering of aspects of that which has passed away.
Memories are implicit and explicit! There is a knowing of the past that is implicit in everything that is recognised.
Just as the blades of a spinning fan look like a seemless whole when they are in rapid movement, conscious-experience also appears like a seemless continuity from one moment to the next.
The rapidity of change from past to present experience contributes to the illusion of sameness.
We require more than the simile of the chariot for a detailed understanding of consciousness - from an early Buddhist perspective.
In order for the visual perception of a 
to arise it takes time for the light that bounces off the object to arrive at our eyes.
The awareness of a is a process that arises sequentially. After seeing and feeling that something has come to our attention we may be able to ‘identify’ what it is we are experiencing. We can get it wrong, like the man who mistook his wife for a hat!
The perception of a requires a memory of what trees are like. What a tree looks like or, sounds like when the wind is blowing etc. Memories are retrieved - a recollection - we live in the presence of the past.
Every new moment is without precedent but our perception of it is yesterday’s news. This is why the mental aggregates are ‘dukkha’ (unsatisfactory).
The mind can only provide us with a semblance of the the truth which liberates. The greatest recipe is not equal to a crust of bread.