My only experience with this is with a lacquer finish. Doesn't hold up if exposed to water.

A little sanding, matching stain, and some polyurethane and you should be fine.

I don’t know anything about repairs, but I just want to comment that the “water damage“ reason seems a bit farfetched to me.

The damage you picture is under the drawers next to the range, right? Several feet from the sink? Does water drip off the countertop there when you wipe it down? I don’t understand where the water is coming from.

That looks more like wear than water damage, and it's kind of in weird spots for water OR wear. The raw spots at the bottoms are really odd unless you mop the floor with a giant ball of sandpaper.

I guess I'd find a matching stain pen and try to hide it.

We have painted rather than stained, but the same layout. We initially hung towels on the cabinet handles, and stopped doing it because the water from hands would catch on the rails and discolor the paint. I would agree that it seems like an odd place for water *unless* that drawer bank either sometimes has a hand towel hanging on it or holds your dishes (as the equivalent drawer bank in our kitchen does). Even when we try otherwise, we get water onto those drawers unloading the dishwasher because we keep our dishes and pots in the two drawers there. Our drawers in that location look similarly worn, but it's because of those two factors. (They're also quite a bit older, but I seem to remember having issues there relatively early on because we were constantly banging things into them/dripping on them.) Also check how/with what you're cleaning them and be sure you're not inadvertently using a cleaning product there that is eating at the finish (or even just sitting on that ledge while you're cleaning, adding moisture).

I meant to respond to you specifically on the other thread but I got busy and it slipped my mind. Is this happening in other places or only on that specific area?

In my opinion, you definitely have a finish problem. Obviously the finish is crosslinked great but it is not adhering to the cabinet very well as it is chipping off in every place where you expect to see when the crosslinking causes the finish to lift, which in turn reduces adhesion. A properly adhered finish shouldn't chip off like that. Water should not cause that much chipping especially on the non-horizontal surfaces. The chipping under your drawer and at the bottom is a sure sign of a larger problem.

Having said that, you might have compounding problems and that makes things a bit tough. If the adhesion is only failing where there has been some moisture then the moisture may be the catalyst that reveals the adhesion problem. In that case, your cabinetmaker is unlikely to resolve the issue in your favor because they can simply say that the issue would not have occurred but for the moisture. While that is true, it ignores the fact that the issue shouldn't have happened even with that level of moisture.

As for my advice... I would accept any resolution they are going to do for free, I would consider resolutions they offer at a significant discount, and would avoid any resolution that approaches retail payment because they are not likely to do it better the next time.