Where the Missouri and Mississippi meet, they each contribute roughly 50% of the water. The Mississippi was better known, so the Missouri became the "tributary" in name.
If the opposite decision had been made, the river would be even longer in the books.
Interesting. I thought it was because the river emptied into the Gulf closest to the state of Mississippi. Like, it would have been slightly unintuitive to have named it the Missouri River all the way down past those Gulf Coast states. The history of river names and deciding which name continues after a confluence is quite interesting.