Just allow zoning of an animal meeting zone. Add it into the regular activity zone, but make the animal meeting zone apply only to animals, separate from the regular meeting zone option. It would be possible to select both options from the activity zone menu if you wanted to which is how current meeting zones work. Or you can create two different meeting zone areas, one for dwarves and one for animals.
This would allow you to create a meeting zone for animals only, and so all animals would hang out in their own designated meeting zone instead of in your dining hall.
Saying to animals 'hey can you hang around here please and not wonder off' generally doesn't work too well. A stile would be good, but pathfinding issues do need to be sorted.
They already do that with meeting zones. This would just split off the meeting zone function of an activity zone into dwarven meeting zones and animal meeting zones. The behavior is already in the game, its just modifying it slightly.
Also this could be very beneficial for fortress planning with the way meeting zones work. For example, create an animal meeting zone just behind the guardhouse, so all intruders must pass through the animal meeting zone to reach the main dwarven meeting zone of your fortress. This means all of those stray dogs, wolves, lions, elephants, bears, and other creatures will hang out there providing a last ditch line of defense to your fortress, and any losses will quickly be replaced with reproduction.
Or create the animal meeting zone in a room surrounded by butcher and tanner workshops for a meat factory.
Animals already flock to meeting zones, so its just taking that current, existing behavior and just splitting the option, so that animals and dwarves don't always need to hang out in the same place.
Assigning animals to burrows will also work and do the same thing. Either way works!