pigs can't walk through open doors. (maybe other animals can't too.)

Confirmed.

I believe it's intended, as they are one block thick, and the door hinge is blocking their path.

But they CAN go through fence gates.

Can confirm. Not sure if intended though...

Cow and sheep can neither.
Wolves, Ocelots, Chickens can pass.

Fence gates, when they are open do not have a collision box.

Edit: Cows and Sheep are fat.

Sometimes it's that two animals try to go through the door (or fence) at the same time and block each other.

They can path through gates just fine on their own, but when being lured by food, they don't always go through the gate.

Pigs are indeed to fat to fit through an open door.