From a direction at the far edge of psychology away from neurology (object-relational theory interpreting group therapeutic experience in the tradition of Tavistock process) we can think about consciousness as a less personal experience.
From that point of view consciousness itself is a shared pool of 'cathexis' (just a term for the mysterious pseudo-energy behind attention that makes things seem more or less important) rather than an individual thread of experience, but our personal memory coalesces our individual sense of our own single consciousness, and that memory resides primarily in a single mind.
So it is impossible to identify 'a consciousness' as being the same or different from another, only a set of memories, and your question loses meaning. The set of memories empowered by a given pool of consciousness can be the same as it was yesterday, or it can be different.
By virtue of being memory, it has some stability, but it also changes every time it is accessed, (whether you attribute that to "complex formation" or Hebbian learning.) So over time it is less and less the same memory. And if something unusual happens that takes the complex of memories apart, it may suddenly constitute multiple separate streams of memory. (E.g. one remembers some things about oneself while sleepwalking, and not others.)
That is my answer, but it needs a lot of framing to make sense to may people. Pardon my long-windedness here. I, personally, just find this fascinating.
Humans communicate, not only explicitly in language, but more generally by doing things that have the intention of being observed. That intention is often unconscious, but it creates patterns that create group identity and assign roles to individuals in interactive situations. In an emergency, for example, say if someone spontaneously collapses in a public place, someone will shortly attend to the body, someone else will move to alert authorities, someone will spontaneously start coordinating the event as 'news', guessing 'What happened?' and hopefully controlling the level of fear, while distracting onlookers from interfering, more and more nuanced informal roles arise in more and more complex situations. These assignments of people to roles are made neither explicitly nor arbitrarily, they are negotiated by unconscious interactions within the crowd.
This is group decision making, and so it constitutes communication. To oversimplify rather drastically, if consciousness is a prerequisite for communication, all of the consciousness that is unconscious to all of those present, yet coordinating all of this communication, must reside somewhere. If you head far enough down this path and just observe, it seems that the vast majority of the consciousness is not assigned to individuals, but goes back and forth between them in an ongoing group process.
At the same time, internal consciousness seems to be driven by the same sort of thing. Different parts of one's experience are 'cathected', pointed at a specific object, and the energy of that thought snowballs or melts away. Different trails of communication result, making up strands of internal conversation like the individual conversations at a party. What integrates a specific one of those strands seems to be the collection of internal referents that the coordinated processes have rendered conscious. Basically, a thread of conscious thought is a semi-organized pile of memories that it has marked as temporarily important, and little more.
So thinking of the individual as a coordinated group process, the consciousness that contains all of these threads is likewise held together by its shared pile of memories, and nothing more. Taking that to its logical conclusion, conscious itself is a shared, ongoing process which resides in no particular thread of considerations. When you are with others, you both maintain your individual consciousness and all share a single consciousness, and when you are alone, you segment yourself into multiple parts that do exactly the same thing. What constitutes a given 'consciousness' here is the pile of memories it curates.
There is shared memory, in that cultural traditions and other experiences may be reconstituted from multiple people in a sort of ceremonial sharing of the experience, but primarily, we have privileged access only to our own individual memories, which are very much stored in our separate brains.