Does the Sole Survivor know how much time has elapsed since the kidnapping?
Yes, later on during the main quest the Sole Survivor does figure out how much time has passed since Shaun was kidnapped.
During the quest Institutionalized, when the Sole Survivor first visits the Institute, they meet the leader of the Institute, Father. It turns out Father is Shaun, and 60 years have passed since he was kidnapped. Up until this time, the Sole Survivor assumes about 10 years has passed since the kidnapping.