This is essentially correct, and it's a bug. If you gain Fire score in the last frame of the previous round (because you're playing Control in Quick Play or because you're in most Competitive game modes) and that made you On Fire or you were already On Fire, then Fire persists to the next round.
Gaining Fire score in the final frame of a round isn't too difficult if you're playing a hero like Mei or Moira and you're dealing continuous damage to enemies as the round ends, or you're on the objective and gaining score for being there.
Unfortunately this one was just rare enough that we didn't catch it internally, but it will be fixed in the mid-season patch *very* soon!
We've got several other fixes and tweaks to Fire coming with that patch that are intended to address many of the complaints mentioned in this thread. Thanks for all the reports and feedback that continue to help us make this rebuilt version of On Fire better!