This is a longtime argument of the Landowner vs. Conservation Agencies, and one reason (of many) that some high fence operators want to have deer and other animals inside their fences put on par with livestock and oversight transferred to their state's agricultural department.

Landowner puts up a fence and says it's his land, he can do whatever he wants whenver he wants even if the deer inside it were once wild or he brings in new ones to breed. Also claims he and anyone else can "hunt" these anytime since they're not wild and are "his" deer (or elk, or whatever).

Conservation Agency says nope, they're a game animal and should-must be regulated with oversight and that belongs to them since deer are game animals with hunting seasons, limits, etc.

This was played out in the Missouri courts the past couple of years. It's been a point of contention in other states. Best I can remember, the courts have ruled for the Conservation agencies.