I don't see why this is true. Just because a thing is difficult to precisely define, doesn't mean it doesn't exist, nor does it mean you can't get closer to it.
I could say that I'm "taking strides towards becoming a better person" and that could still have meaning, despite the lack of any objective measure and, even, with the understanding that "a better person" could change over time.
I would argue that "taking strides towards becoming a better person" is a totally meaningless statement. If I were to ask you "how so" then you would probably respond by listing definitive actions that demonstrate attributes of goodness, like generosity or kindness.
I disagree. You could also characterize it as, for instance, having better knowledge of yourself (which, interestingly, seems to be a characteristic of consciousness as well).
In philosophy of mind, self-consciousness is separate from qualia, although some might argue that to be self-conscious, a being must have qualia. Qualia meaning what it's like to be in some mental state. Seeing a color has a color quale, which means there is an experience of color. Or better yet, we experience a colored-in world when we see (or dream, remember, visualize). Does any current AI have some mental experience?
For us, there's something it's like to be self-conscious, just like there's some experience that goes with our thoughts, such as inner dialog (hearing your thoughts). For some, that might be mental imagery instead of internal sound.
At any rate, it's the various feels that make up our experience of being a self in the world, or a self reflecting on itself.
Honestly I think this is largely because humans want the conclusion that only they are consious, or at least that only they and some animals are, and are trying really hard to carve out a definition that makes that work. Why would we do that? Well obviously because we create AI to do work for us and we don't want to have any ethical considerations get in the way of treating our new slaves however we want.
I could say that I'm "taking strides towards becoming a better person" and that could still have meaning, despite the lack of any objective measure and, even, with the understanding that "a better person" could change over time.