Thanks for the fixed link! Yeah the shelves and walls of the cupboards are all some sort of melamine, but the doors and facing trim is all solid wood. That's what the hinges are attached to - the solid wood trim. There seem to be a couple staples and such attaching some of the trim, but not enough to hold them as firmly as they are.

I think they were maybe-built by a previous occupant - a lot of the stuff in the house is high end, very competent, home handyman stuff.