I used some plastic wood filler to fill a hole in the floor. It's noticeable but I wasn't terribly concerned with that and thought I'd eventually find a marker. But when sanding the area after applying the plastic wood, I was way too rough. Can I just spot re-finish this? How do I figure out what finish has been used for this floor?

https://i.redd.it/kgnm0e8196t61.jpg

This floor looks like it has a film type finish. Most floor mechanics in the industry use oil based poly-urethane to get this look. I would buy the smalle3st can of oil based poly and follow the directions on the can for good results. With a small patch like this we follow the grain pattern of the wood surface to apply finish. I think you would find the best results with semi-gloss.
Tape off the area so no-one steps on the wet finish. Apply two coats minimum.