Which Wood Stain & Polish?

The finish on both is failing. It's too late for polish. It won't completely fix it but it would make it look better and keep it from deteriorating to put a couple coats of finish over the top. I wouldn't clean with soap and water, the water is causing the damage. Instead I would clean it with a wax and grease remover. Most auto parts stores which sell paint will have it. You could certainly get it at a place that sells automotive paint. The solvent will loosen the majority of any wax or polish which may be on the wood and the rag will pick it up. Do it several times changing rags often. From there scuff sand the wood with 220 grit or finer paper and wipe off the dust you created.

The handrail is probably finish with a fast dry polyurethane. It's available in rattle can and would do fine for the handrail. The table top can't be done with a rattle can. If you have an air compressor a cheap paint sprayer from Harbor Freight would spray the table. The table is likely done with lacquer and is very easy to work with. Assuming you have never sprayed a finish get some scrap or a piece of cardboard and practice spraying lacquer. The lacquer will need to be thinned and you will have to tinker with it to determine how much thinner. It should apply wet. Visually think of laying roofing shingles. It will spray a path of finish about 6" wide. You start at one end and apply the 6" and the next row move over 3" overlaping the first path like shingles overlap. Don't stop with the finish until you at least reach the middle of the table where the two halves go together. It needs to be a continuous sheet of lacquer. If you start and stop on a section it will have to be sprayed again, a spot will show where you stopped.

Working with lacquer keep an eye on humidity. At 70% humidity or higher water literally gets in the finish and makes it look cloudy. Also if you encounter the finish doing something odd like it wrinkling up stop where you are. Often furniture polish gets into the finish to the point it can't be cleaned off. If you have that problem there is a fix but takes more supplies.