It's a pretty simple answer, if you have two tags, then you can shoot 2 birds, it doesn't matter if it takes 1 shot or 10, you're only allowed to harvest those two birds.

On the other hand, if you only have ONE tag, then no, you can't legally harvest TWO birds, not even with one shot.

In Kansas, it's illegal to harvest more than your tag allotment, even on accident... 3 examples:

My cousin (age 7) was turkey hunting and took a shot right as his target bird came out from behind a bush. Unfortunately, he also killed 2 other birds (a jake and a hen) still behind the bush. Not wanting to waste the game, my uncle called our local "ranger" and explained what happened. The ranger luckily allowed them to purchase tags for the birds, and issued a citation for hunting without appropriate permits, for which he waived the fine (basically to document the event). My uncle and cousin both had to purchase one tag in their name each, because my cousin only had 1 tag left available to him, so my uncle had to cover the 3rd bird out of his annual tag allowance.

I dropped a 2-for-1 buck and doe once. In the mist and the tall grass, I hadn't noticed the doe walk up behind the buck, and one shot dropped both deer. Luckily, I had tags for both a buck and a doe, otherwise I'd have been in a pickle.

I was guiding a friend's son once, and he dropped two doe with one shot. We contacted the local ranger, but the shooter was maxed out on tags, so he wasn't allowed to purchase a tag (as was I). The ranger waived the fine/citation, but confiscated both deer and his tag as informal punishment.

Yes, there's always the option to leave the other animals lay, or maybe even try to sneak it home without tagging it, but at the end of the day, your own ethics are up to you.

Ultimately, YOU are responsible for identifying your target, and everything within the path of your shot. Accidents DO happen, but if the DNR/DWP made concessions for everyone that claimed they bagged multiple animals with one shot, we'd be looking at a never ending cycle of liars, theives, and poachers...