Filling the divots doesn’t seem like an option, the material you add will be softer than the rest of the face.
Clean off the face with a wire brush (rough, medium, then soft) then brush the rest down with a very fine scotchbrite pad and oil.
I used this process for my Peter Wright when I got it a few years back. Took about 4-5hr, I went very slow during this cleaning, I didn’t want to remove too much material. Mine was in fairly nice condition when I got it, but if you just take it slow you should be able to clean yours up no problem. The horn of you anvil is is much nicer shape then mine, I’m a little jealous hahahahaha. Someone made a bunch of large cut marks in the horn on mine.
I don’t see any way of telling the maker of your anvil. Just check to make sure it bounces a hammer (or a ball barring) on its face. Also be sure to check the all the faces of the anvil for cracks. When I bought mine there were 2 at the antique store I got it from. The first one I saw was bigger and had less rust, but after flipping it over I noticed a long fine crack running along the bottom of the anvil. So I skipped out on that larger one, never know if a crack like that will continue to grow.