I don't actually think this behaviour is a big deal.
My DD is like this (she has anxiety FYI). If she shows me something, I respond to her factually and casually. I have read that you shouldn't tell them their feelings are wrong ie "oh come on it's not that bad". Their feelings are valid and have every right to have them, and have every right to come and tell their parent.
So for eg if DD comes and tells me she has just bumped her elbow, I will say "ooh I hate it when that happens, well I know it hurts now but the pain should start fading soon".
Or for a scratch that I already know about "oh how's that scratch going, it looks like it's healing well".
Just respond and move on. Easy fixed. All they want is some reassurance.