Are there any dishes that can only be cooked with a microwave?

Yes!

In 1969, the physicist Nicholas Kurti gave a talk in which he demonstrated a variant of Baked Alaska called "Frozen Florida": a shell of frozen meringue around a center of hot liquor. This was done by chilling the meringue and the liquor together, then cooking in a microwave oven which had a rotating platter and no stirring fan. Because the microwave beam was always heating the center but only intermittently heating any given part of the meringue, and because the meringue was low-density and frozen, the liquor could be heated while the meringue remained frozen.

(The idea was broadly similar to, and presumably inspired by, radiation therapy for cancer, in which a tightly focused beam of radiation revolves around a particular spot which may be deep inside the subject's body: only that single spot is always being energized by the beam, so it receives much more energy than the rest of the body does.)

The "instant" sponge cake, innovated at El Bulli, can only be made in a microwave. Here is an example. Basically, a batter is poured into a whipping siphon. It is charged. The aerated batter is dispensed into paper cups. The cake is cooked in a microwave. The cups are removed and inverted. The cake is released. It is easy and fun to do.

At least for smallish and thin objects, microwaves heat foods “everywhere and throughout at the same time” (for lack of a better description). Exactly what isn’t desired for a steak, as discussed in the Q/A that inspired your question.

To achieve similar results of heat distribution with other methods, you either

• stir the food (for cooking in a pot)
• supply heat from all sides (when baking in an oven or steaming)

If both don’t work, because you don’t want to heat from the the bottom or surroundings only and don’t want to stir, your can only use the microwave:

Place a Schokokuss for a few seconds in a microwave to puff it up. Unlike plain (and firmer) marshmallows, they become soft, gooey and semi-liquid. Total guilty pleasure and comfort food, but can only be made with a microwave. Use a low setting and take it out before it disintegrates completely. Here’s a random video (in German, but the text is utterly irrelevant).

I saw someone point out the El Bulli instant sponge cake, and it immediately reminded me of a different recipe: The Microwave Chocolate Mug Cake. I know it's not exactly a microwave-only recipe since cakes have been around since antiquity, but this is a specific recipe for the microwave with a couple advantages over an oven baked cake.

A microwave cake is done really quickly: it's essentially just mixing a bunch of ingredients in a bowl, pouring the batter into a mug, and putting it in the microwave for a little over a minute. The result is a surprisingly spongey cake that can be made even by the most inexperienced of chefs in per-person servings, instead of having to make an entire cake that you need to finish in a couple days.. there also is a lot less cleanup to do after this recipe: a mug, a stirring implement and a bowl, compared to 3 separate bowls for the egg components and the cake batter, something to whisk the eggs, something to mix the batter and a cake mould. I have personally made microwave cakes on occassion and it was surprisingly good.

If you nuke an onion half, with an X cut in the middle as deep as possible but not through the outer layer, and seasoned with lemon pepper and a pat of butter on top, until the onion is soft, you have a tasty serving of vegetable

I don't think any other way of preparing an onion would taste less "oniony".

Possible duplicate of this question. As mentioned there, there is the intriguing idea of a Vauquelin - a heat-stabilized egg-white foam, somewhere in between ice cream and meringue.