When fishing an entity, the bobber's position will remain at the point where that entity was first hooked. This can be seen when the entity travels over 32 blocks from this point (the line will break) or is killed (the bobber will drop at that location). I expected the initial hooking point to be forgotten, instead moving along with the entity and mobile bobber.
Tested in creative and survival mode; no available option appears to have any effect on the issue. On an LAN server, the bobber will visibly be stuck in place at the hooking point, the texture without any reference to the false position.