Most certainly go through the same thing. MOST of my worries are still directed at myself, but I also get intensely anxious about the health of my parents and brother.
I think it's relatively normal for anxiety to shift around. In the same way that we often jump from one disease to the next, I think a lot of us also experience our anxieties shifting from one person to another.
I try to remember that we are each our own person and my worrying about my mom isn't going to change a thing. If anything were to happen to her, it would be devastating, but I could handle it. I am stronger than I think I am and I guarantee you are too.