Advice on table top repair needed

A friend has asked me to look at the tops of a side table and coffee table, both identical in style only the size differs.
The side table has had indoor plants on it at some stage and consequently there is some water damage that has caused the finish (varnish?) to lift and crack. It seems to me that the only solution is to sand it back and re-seal. I feel sure that the top is veneered as the timber underneath looks similar but is not quite the same. Any thoughts on the best way to tackle this? Is it possible to match the finish so that I don't have to sand the sides as well.
For the coffee table the top finish is sound but there are a number of white rings on it. I have some Howards Restor-A-Finish which is claimed to do such work.
Thanks
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Definitely veneered methinks, based on the direction of the grain on the edge. I’d very carefully sand it back to bare timber. Paint stripper might work, but I don’t know whether it would cause the veneering glue to breakdown. Hopefully someone else knows the answer to that question.