And there were a couple of wars before those. They happened so frequently and quickly, was there really no reason to not call it one continuous war?

2 reasons: 1 there were actual breaks between coalition wars and the peace of Amiens in 1802 provided a real, short truce from all major wars same with the gap between the 6th and 7th coalitions (weak argument) 2. while there are some problems i have with the specific naming of the coalitions (the fourth coalition is "really" just prussia joining the third coalition late after the crushing defeat at Austerlitz which knocked austria out of the war soon after) they do highlight some important diplomatic points.

Given how dominant France was her numerous campaigns saw (with the exception of Britain) pretty much every state in Europe compelled to support him at one point or another while facing a changing cast of enemies at each specific time. Splitting "the napoleonic" or napoleonic/revolutionary wars into coalition wars lets us separate the "wars" out by diplomatic relations (e.g. 3rd coalition involves "the third coalition created against a revisionist france").

I would argue we actually do sort of view it as one continuous war also: "the Napoleonic Wars" but by splitting them/it up into multiple coalition wars we have a better sense of who is fighting who over what.

Anything in this post that i can clarify?