Highly likely it's a lacquer finish and highly unlikely it's polyurethane. Poly is just not a production finish.

It's hard to give advice without seeing to and from finishes.

Stains are really designed to be applied to bare wood. There is one exception and that is when you are using a gel stain as a glaze. The glaze is applied between coats of finish and you can use a glaze to "dirty up" or slightly tweak the color. Glazes need to be top coated after dry (24 hours, usually). You don't have much to loose by trying this approach. Buy a can of gel stain and a can or aerosol lacquer and give it at try. Just don't put the glaze on too thick or the top coat won't bond. You can repeat a couple of times.

Another option is to apply a toner. This is a finish (usually lacquer) with color in it. You can buy these in aerosol form at places that cater to professional finishers and touch up people. Mohawk is a common brand. Just apply in light coats and sneak up on the color, because it can get too dark or opaque all at once.

Worst case, these don't work and you'll have to strip and refinish anyway.

But do use a chemical stripper, don't rely on sanding off all the old finish as this is a terrible way to do it.

Here is a link that might be useful: start around step 5 here.

OK, thanks for the information, Bobsmyuncle. I'll look into both options. And, yes, I've stripped paint off wood with chemical strippers and it works very well. -- glad you reminded me of that before I broke out the sand paper!