This probably depends a lot on the size of the studio, the size of the animation crew, and the diversity of experience levels on that crew. The smaller the studio, the less options available to the supervisor, and the more likely it is that the animators may be thought of as interchangeable artists.
However, you ask about the larger studios, and you're right that it's treated differently.
The larger studios have the luxury of a larger animation crew, probably with a somewhat diverse set of backgrounds and specialties. The process that you're asking about (handing out shots) is often referred to as "casting" at these types of studios, and that's exactly how many of these studios approach the process.
They often view the assignment of animation shots or characters as "casting" those animators in the role that they are most appropriate for. Someone who's shown great comedic timing in the past may be cast to work on a lot of shots that feature comedy relief, for example. Or someone who's shown that they have amazing body mechanics might get cast on a lot of fight scenes.
However, many other factors come into play with casting, and it's rarely as defined and simple as I laid out above. While the goal for any of these larger studios is to use their crew as efficiently as possible, many other factors are considered. Among these would be:
- What the animator is most passionate to work on
- What the schedule and budget allows or forces
- Scene difficulty level vs. animator skill level
- What the surrounding shots are
- Crunch time
These are all important factors, and in any good studio, all would play a role to some degree in the casting of shots.
Near the end of a project, though, crunch time and deadlines increasingly become the deciding factor in crewing a shot or sequence. The project has a deadline, and when it really gets into the final weeks or months of a project, the most important deciding factor in casting becomes availability. Who is free to work on this? If the Supervisor feels that their talent level is sufficient, and they are done with their other shots, they'll probably get thrown onto whatever shot is next in line, regardless of any of the other considerations.
This is totally normal. Many projects at many studios start out by casting long strings of sequential shots to one animator, but by the end, it's natural for the schedule to force a more scatter-shot approach in order to hit the looming deadline.