how do I identify what the finish is that was used here, so that I can identify the proper repair?
There are some test you can perform to narrow down or identify the finish used on something. Sometimes a good educated guess can be right on the money, pieces from the right eras will inevitably be finished in only certain things (shellac before a certain date, lacquer after until recent times).
The presence of MDF tells us this is a modern piece so there are a few possibilities for the finish, but what was originally used doesn't really matter because repairs don't need to be done in the same material, in fact they frequently aren't. It's quite common in refinishing for the colouring of spot repairs to be done in a different way to originally, and then for a different final finish to be used.
Hopefully the repair can be localized, and not require resurfacing/refinishing the entire bar.
Absolutely. In the worst-case scenario you'd be looking at refinishing the top only, not the entire cabinet.
But spot repairs are possible and this is definitely what I would try here first.
It looks like you need to:
- Build the lost material back up.
- Restore the colour.
- Equalise the finish.
Repairing any lost material is fairly easy at heart, you can use a range of things both commercial and homemade. These include a basic wood filler, polyester body filler (e.g. Bondo) and sawdust/sanding dust bound with epoxy or PVA glue (white or yellow).
General practice is to slight overfill, then when the filler material has set hard you flush or fair the repair. This can be done entirely by sanding, although there are other methods for removing the bulk of the excess (which I prefer to use myself as much as possible) here I think sanding is the ideal method due to the shape involved. However, the damaged area is a compound curve so matching the curvature of the repair will be more challenging than it would be otherwise and great care will be needed not to create a flat spot (or more than one!) which will be noticeable after the finish goes on.
To some extent the second two goals can be done together, e.g. using the right colour of shellac, but they are often tackled separately. Doing this separately is what I would advise here, although some final toning of this corner may (I think is likely to) prove necessary to hide the repair as much as you'd prefer.
This is obviously just a brief description of the processes needed and not a full how-to. I strongly recommend looking at a number of restoration vids on YouTube1 AND reading at least one good, modern, book on restoration before going ahead here2.
1 Preferably hosted by full-time professional restorers, not some random dude/chick working in their garage or driveway. While the latter are not all bad by any means there's no telling how questionable their practices might be, e.g. using waterbased products exclusively because they "prefer not to deal with solvent odours", and regardless of whether they're the most suitable to the task at hand.
2 There are plenty of slightly dated and historical books on furniture repair and restoration, and while some of the processes and methods used are applicable to today many are not. So for now it's best to focus on sources where you can do exactly as described, and there's a better chance they're using products you can still get today.