The problem seems to be quite widespread with many unhappy players taking to the Steamworks forum to vent their frustrations and seek solace. At the time of writing, there were over 500 comments in a 35 page thread.
Ubisoft is aware of the issue and has been using the official Watch Dogs Twitter account to communicate with players. It said that the sign-in system might work after a couple of tries, but that the team is working to fix the authentication issue affecting online services. Apparently, there are signs of improvements.
Affected players are being asked to stay tuned to the Watch Dogs Twitter account for updates.