Cats definitely empathise with their ownerws - or 'pack leaders' as they are perceived.
Illness and stress give of vibes which an animal can sense, and it does make them uneasy, so trhey tend to watch and see if you are going to be OK, so trhat everything can return to normal.
It is endearing, but remember, in a 'wild' situation, a sign of illness in the leader would be seen as a sign of weakness by the upcoming males, and a potential change at the top which will affect the females, so your cat's concern may be an instinct for self-preservation as much as concern for your welfare!