This looks like solvent peeled up the urethane. I don't think you will have great success coloring it.
The best way to do this is to re-finish the entire top of the table. It's not as bad as it sounds, but if you haven't done anything like that before, then get a professional to do it. Experience is necessary.
The first thing I would do is sand the rings and use a little oil based polyurethane to "fill-in" the rings. If the color matches (it should), then lightly sand the area (make it all level). Then lightly sand the whole surface with 220 grit paper (not to remove the varnish, just prep it for a new coat). Keeping the old layer of varnish is usually better for making a smoother finish. Tape the unsanded edges and protect the other areas with news paper or plastic and recoat the (entire) top of the table with urethane. A spray-on clear coat might work.
If the oil based polyurethane did not match the color (the rings were lighter or darker) then you will need to sand off the old layer of varnish (on the top) and then refinish it. Matching the old varnish (on the sides and legs) may or may not be important. If the color is close and only different where it changes from the top to the sides, it will probably not be noticed.