The distant points are mainly galaxies. Stars in our galaxy usually are very bright and have “spikes” of light due to optical distortions. There are probably billions of galaxies. I know that there are organized efforts to categorize galaxy types by volunteers. There is training on galaxy types for these volunteers, but I don’t know that there is any attempt to name them.
I sure wouldn’t want that job, LOL.
I think you are saying that virtually all of the small dots in this picture
( https://apod.nasa.gov/apod/ap220902.html ) are galaxies? And that not even a dozen are stars in the Milky Way?
Some of those objects must be a long way off.