I mean "same person" only in the sense of your consciousness, not in terms of your knowledge or personality or anything physical. I'm not sure that there's some "self" experiencing one's consciousness over their lifetime or that there's anything linking ourselves between different points in time other than our physical bodies and our memories.
I think our conscious lives might be just a collection of moments created by our physical mind, and that there is no underlying self experiencing all the moments, meaning that the consciousness experiencing your life at this moment is not the same as the consciousness experiencing your life 10 minutes from now any more than it's the same one experiencing someone elses life right now.
It's not a question that bothers me, I just find it interesting. But I do think it's relevant to real-life. In terms of selflessness, if my consciousness in the future is no different than that of another person, then I should be just as concerned about their well-being as I am for my own in the future. I think you should care about other people's well-being regardless, but this is another way of looking at it. Also, if my consciousness in the future is no more different than that of someone else after I die, wouldn't that be a type of reincarnation?