It's not a classic technique; but relatively new. It's sometimes called "speed basting" in US kitchens.
FWIW, the actual cooking of the fish is, in "classic" terms, somewhere between a sear and a saute. If you were sauteing it you'd be tossing it or turning it a lot. On the other hand, you use more fat than you would in a sear. You can do other proteins in the same way, but they have to be relatively quick-cooking.
It works especially well with fish, because you can cook skin side down to get the skin crisp, while not ignoring the top; keeping the rarest, slightly translucent "center" close to the actual center.
You can't leave the pan on the grate, because you won't build enough of a reservoir of fat by tilting the pan to baste -- that is, if you don't use too much fat. Because the pan is off the grate, you need beaucoup flame to do a good job. And don't forget to tilt the pan both ways, so the fish cooks evenly. Even good cooks have a tendency to just do it one way.
A whole, peeled garlic of clove and aprig of thyme or rosemary in the pan are "classic," for speed basting as well. You can discard them or use the sprig for garnish.
I generally discard the butter for cooking, and use the hot pan and whatever fond is in it to make a pan reduction (using fresh butter), a caperatti, or a play on beurre noisette (several pieces of cold butter, all at once into a hot pan -- when the butter is sizzling and almost melted, add just a couple of splashes of vinegar and it will go noisette instantly).
Speed basting is a fantastic technique for several things; and better than fantastic even for skin-on fish.