...the one who made the final shot will get the territory.

Restrisiko is correct. Remember there is a 60-minute combat cycle for all ground and naval units, i.e. damage inflicted and suffered is updated every 60 minutes (but every 15 minutes for patrolling aircraft units). So, whichever competing attacker inflicts the final blow on the defending unit (i.e. destroys the defender's last hit point) wins the province. This may or may not be the strongest attacking stack.

Also, BEWARE: unless the two attacking formations from different countries are formal allies or have a mutual right of way, the loser in the contest to capture the province will immediately begin attacking the stack that just captured the province (it has no right to be there by virtue of a right of way or alliance), and you may find yourself at war without intending to be at war!

If you are tricky and time your units correctly you have a better chance of taking the territory I think. I had a coalition member that reached a enemy province ahead of me. I was able to force march a couple tanks into the province center and also hit/kill the remaining enemy unit with artillery just before they reached the center, while my coalition member's stack was still counting down the "attacking in . . ." pop up message. I got the province. We also left that province behind in our march and at day change it revolted and joined my coalition member. Sigh.