Hard to see shade in picture what tone is it

It's a shade of light blue w/ heavy wearing & distressing

Is this a painted and distressed finish? Or a specialty stain?

painted & distressed

Please help! The former owners of our home had painted our kitchen cabinets a dark brown, originally white. Almost immediately, the paint came off anywhere there was any contact. I have done touch ups repeatedly, sanded, cleaned and repainted with kitchen paint but it continues to come off. If I run a fingernail along, the paint just peels right off. I would hate to repaint as there are a lot of cabinets. Any advice? Strangely, all the newly painted surfaces are doing the same thing. Especially the stair railing, the moulding around windows. I have matched the paint colour and bought good quality paint, not cheap, for touch ups from Home Depot and Home Hardware.

Was the original paint oil-based? You cannot paint latex over oil, or it will do exactly what you're describing. Oil over oil, or oil over latex only.