Before you consider stripping it (which is certainly an option) try cleaning it with Lemon Oil and 0000 steel wool. Then you will know what the real problems are. The legs look like the finish is "pickling" - possibly from being near a heat source. Try cleaning it first, then reassess.

Thank you so much I will clean it with the lemon oil and see how it looks. If it cleans up nice should I do anything else to it to protect it?

NO lemon oil!!.... begin with mineral spirits.....which will just remove greasy dirt.
You are probably going to have to do an "easy strip"...that is brush on and wipe off a mixture of denatured alcohol ( shellac thinner) and lacquer thinner 9 acetone)...2/3 to 1/3...it will soften the old checked finish and smooth it out and you wipe off the excess.
Lemon oil is mineral oil with a lemon scent....won't do anything but make your piece greasy. Won't "sink in" won't "feed the finish" will soften oily dirt....but will leave an oily residue.

The rough finish is caused by the oils outgassing from what likely is a varnish. all the soap and water won't change that.
I have that piano stool's twin in my basement. makes a very good piano stool for a short person. It was what I used as a child.
And the metal on the claw legs isn't brass....so don't waste your time trying to shine it up.

Ok thank you.

I have a similar stool that I want to refinish. Are the feet supposed to be so dark?

Do You know what the finish is on it. we have the same stool in fairly good condition but some hand sanitizer was spilled on it and removed the finish. do you have any ideas.

I'm 99.9% certain that is a shellac finish.

The finish is shellac. Age causes that bumpy look since it is a very soft finish and tends to soften, get dirt in it then shrink. Best to strip it all back as you will never remove it all with just soap and water or anything else.