Love this thread (although I think the general tone towards Thecard has been undeservedly harsh). Wish I'd got around to it sooner.
The whole identity/continuity/consciousness question is something I've thought about a lot for at least 10 years. I used to be in the camp that believed a copy of me is not me, and it was all about maintaining stream of consciousness. I find myself kind of drifting away from that recently.
First, the arguments about how we naturally evolve over time or go through rapid changes and are never the same person we were in the past have really sunk in. I think that's one aspect of the subject that just takes a lot of time and experience to fully process. That's where me 10 years ago would have re-asserted "but that's why it's the continuity that matters and not the consistency". I still think there's something to that, but I don't necessarily think a break in continuity is the same as death anymore. It's something undesirable, but... it's... actually really difficult to describe my position on it, anymore. I think there are multiple facets to identity. It's not a binary thing, where you're either dead or not. Loss of one facet isn't loss of the whole.
I see two major types of identity that are related and interact, but are not quite the same. There's what you experience as your identity, which I'm increasingly recognizing as incredibly flawed, and your objective identity as could be described by an separate observer. Experiential identity hardly needs any description. It's what most of the discussion here has been about. All I'll say is that it's a messy patchwork thing. A frankenstein collection of mostly incomplete moments, notions, and falsehoods that is constantly evolving, often adjusting to match whatever our ego's current styles of thought and perception happen to be. One of my biggest influences in truly understanding this was a single sequence of panels from the fantastic graphic novel Asterios Polyp.
So... where is identity in this? You can say that it's in the moment. Our collective experiences and innate tendencies combine to make us who we are right now. But how much of us is really here right now, and how much of you is locked up in constant reflection and processing of the past, altering your perception of yourself and past/present/future realities, or cultivating future identities? How can our identity be in the moment, when part of us is always examining other moments?
I don't mean to belittle that part of us, either. I think it's really important. It just isn't enough to completely define a person. It seems to me like our cognitive processes are more central to our experiential identities than our memories or the hardware.
And that sort of brings me to the other major type of identity, which is a very strange alliance between my natural western enculturation in egoism and my loosely informed eastern spiritual influences. The best way I can describe it is that the other part of our identity is a memetic entity. We're an idea in action.
The best thing I've been able to liken our identities to is the weather. There are a lot of general consistencies, that are still subject to large degrees of variation and change, but the details are never the same. If your life was made into a map, there'd be other similarities. There would be The Weather as a very general term, indicating things like rain, sunshine, etc as univerals (your most prominent personality features that remain relatively constant). The weather in any region also has its own personality (your stages of youth, your goth phase, how you were when you worked at that one job, etc). And special events, like hurricanes, are generally treated as distinct entities as well (the only time in your life that you got really drunk and weren't at all yourself). Imagine two strangers from completely different parts of the world talking at an airport about the weather. They'd be talking about similar features, but in completely different contexts with many notable differences. This is the same as a person who knew you 15 years ago talking about you with someone who knows you today.
I think I can take a hit to the identity that I experience in order to keep my memetic identity alive. I'd be ok with being copied, which would keep me alive "in spirit". Sort of like a piece of software, which is an analogy I think I've seen in this thread already. When you're copied, you're not creating You 2.0. I think we can all agree that Doom 3 was nothing like Doom 2, but fortunately Doom 2 is still alive in spirit. After being ported to new platforms and engine upgraded and built-upon in many ways, it's still Doom 2. The game lives on. If those things didn't happen, it would be dead. Similarly, I would still consider myself alive so long as my memetic identity is able to manifest in the world with individual agency. Complete survival of this specific manifestation of myself would be the most preferable thing, but I wouldn't consider loss of it to be a complete loss of myself.