No..it is the reflection of the blood vessels in the eyes. When the iris of the eye is opened wider in a dim room, when you use flash, the light from the flash penetrate the eye's opening/iris and hit the back of the eye where your retina is located along with the massive amount of blood veins there..so it reflect the blood red colour forward back out through the eye/iris and that redness is capture by your camera.
That is why some camera comes with red eye eliminator function. It fires off 1-2 flash in advance of firing the main flash to capture the shot. This "advance " flash will cause your subject's iris to close down to a smaller opening ( this happen very fast ) so that when you are capturing the actual shot, your iris would have become so much slower that it makes it hard for the red blood cover to reflect back out and capture by your camera.