Doing It for the Art: A reason for the rotating directors, so each film would be stylistically unique from the other entries. Even when Christopher McQuarrie ended up being the first director to return for a sequel, he was instructed by Tom Cruise to change his direction style.
Reality Subtext: The strong chemistry between Ethan Hunt and Ilsa Faust is helped considerably by the fact that Tom Cruise and Rebecca Ferguson, who play them, have a very warm and friendly relationship in real life.
Recycled Script: Both the first and third movie involve Ethan being set up by a mole in IMF who isn't revealed until near the end, and who tries to frame the boss of the organization who spends time as an Anti-Villain trying to hunt down Hunt, with the audience left unclear if its because the boss really thinks Hunt is a traitor or because the boss is the traitor. In both their actions force Ethan to go rogue and steal the MacGuffin that the bad guys are after for them, and in both The Mole is in league with an arms dealer. Each movie sees Hunt's family being dragged into the plot to get at him, sees a fairly traditional IMF plot go off without a hitch only for the team to be ambushed after the fact, and features a reference ot Ethan evading airport security despite IMF putting "a guy at the airport".
Sequel Gap: The series constantly flirts with this, owing to the busy schedule of franchise lynchpin Tom Cruise. Of late the gap has been gradually narrowing, going from a 6 year pause between II and III, to 5 between III and Ghost Protocol, then 4 until Rogue Nation, and now 3 years into Fallout. The seventh and eight films are scheduled for 2021 and 2022, which means they're possibly a case of Movie Multipack.