It's been ages since I've left gamedev, but generally characters and creatures are animated by constructing an actual skeletal structure inside their body (usually called a mesh). Points on the mesh are then mapped with various weights to the bones in the skeleton. When a bone moves, points on the body are moved along with it depending on the weighting. You can think of it like attaching muscles to bones, which determines how the "flesh" moves when bones move/rotate.

What you animate then is not the "flesh", but the skeleton. You tell the hand to move to a certain position, and then a constrained optimization problem is solved to extend the arm and hand while keeping the joint limits in mind.

The most convenient default shape to set up the skeletal structure of a humanoid is the T-pose, so this is usually the shape humanoids take in games before any animation logic is applied to them. If you see a T-pose in a game, odds are that the animation system crapped out. The animation system may have crashed but the error was swallowed to keep the game chugging along.