Is growing bonsai a special case of hydroponics?

Bonsai are generally planted in soil that's specifically blended for the particular sort of bonsai species. Bonsai soils of the same type may be packaged in different sizes of granules, eg 0.5cm or 1.5cm etc, and some bonsai soils aren't granular: they have different compositions etc. The same type of soil may have different Romanisations of the original name. But they are soils. Often the soil for a tree will be a blending of several of the specialised soils, cf simply using one kind only, and at the bottom something with a coarser consistency for drainage & aereation.

They are small trees, and their environments are small, yet complete. There may be some specimens which are grown hydroponically, but generally bonsai are grown in soil, soil which often consists of of a blend of more than one specialised bonsai soil component, and with some coarser soil/ gravel at the bottom. Also, some trees simply don't do well unless in well drained soil. Often the soil would be different from vegetable soil, but might contain eg 25% vegetable soil in the blend. Also, bonsai soil needs to be replaced every several years so that the soil will continue to be able to provide micronutrients which the bonsai draw from the soil. Bonsai generally aren't special cases of hydroponics.

In a sense, yes, bonsai is almost hydroponics. I grow my bonsai in Turface MVP, a totally inorganic, high-fired montmorillonite clay that is marketed to sop up water on athletic fields. It is like coarse sand or gravel but retains much more moisture than either of those materials. In the opinion of many, the 'ultimate' mix is grains of lava, pumice, and a clay from Japan (Akadama) that is added for moisture retention.

One supplies NPK and minor elements via fertilizer. Some use liquids applied during watering, such as Miracid, and for these it is exactly as in hydroponics. I use Osmocote-Plus prills that release their contents, depending on temperature, to roughly match plant needs. So maybe this is even closer to hydroponics with a high-tech control system.

Roots require oxygen to live and grow. These bonsai substrates have high air-filled porosity. Roots also must be kept damp, of course. These substrates/mixes also simultaneously have high moisture retention. Wood chips and/or chipped tree bark are the best organic substrates, but they do break down and the air-filled porosity correspondingly reduces over time. These hard, inorganic substrates do not. Conventional potting soils also compact and cannot achieve the same levels of air-filled porosity and, hence, they become highly problematic in shallow pots.

One thing these hard inorganic substrates do that hydroponics and soft soils do not is produce root ramification. When growing root tips deflect around hard particles, they branch or ramify. Plants in these inorganic substrates develop densely ramified root pads that are highly efficient. The epidermal cell wall extensions called 'hair roots' occur a few millimeters behind the growing tip, so more tips = more water and mineral adsorption. Conventional hydroponics and growth in soils are missing this aspect.

I use normal potting soil for my bonsai, so I would say no it is not a specialized form of hydroponics.

What's important with bonsai substrate is that it has good drainage. If water stays too long in the soil/substrate, you'll get root rot which is obviously not good for the plant.

The bonsai soil you are referring to contains a lot of grit or rocks, which have good drainage capacity. However, with this soil you'll have to feed them regularly with nutrients. Like almost every time you water. So in that way it has similarities with hydroponics.

For me, with normal potting soil however, I don't need to give them nutrients so often, maybe once or twice a month during this time of year.