The Giants’ draft-day trade for Ole Miss quarterback Eli Manning was the first major headline of the Tom Coughlin era, which had begun just three months earlier in January 2004. So it would be somehow fitting if now, in December 2014, both men were entering their final month with the team.

They have always been inextricably linked in the minds of football fans, who remember Coughlin and his staff developing Manning from a bumbling rookie into a two-time Super Bowl MVP. Making a joint exit seems like the proper thing to do,...