All cabinets will experience normal wear patterns and chipping at those outside edges that you show. It just shows more on painted cabinets. And it usually happens over time rather than in a short period of time. That’s what they should look like after 5 years. Not 1 month.

Cabinets shouldn’t be painted on site, or with house paint. Factories use conversion varnish in clean rooms for a reason. It gives a high quality tough durable finish. That is what any “custom” cabinet maker should also provide. If they didn’t? They are not likely to be true custom cabinet makers.

Custom is not synonymous with quality. It just means created for you. That can happen with cheap low quality flat pack that get slightly field altered. People think it means something that it doesn’t. The devil is in the details. What were the specified details?

"The builder reports he used high quality Sherwin Williams paint."

How was it applied? What type of wood? Site finished I assume? Prep of the wood? Sealer is not stopping that and not typically used over painted surfaces.

It was built on site. I can ask what type of prep was done, but the builder is indicating this is normal. Any ideas on how to correct this?

This practice of cheap site building and finishing with house paint only exists in TX for some reason. The basic issue is you picked the wrong contractor. You can’t overcome that innate deficiency of processes. Pick better tradesmen next time. And have him show you how to touch that up with the house paint. It will be a frequent occurrence on your part to have to deal with.

It would help to know exactly what finish was applied to the island -- primer and topcoat.

And what material or materials is the island made from? Type of wood etc.

Any pictures of it being built?