A basement under a double garage is area-wise equivalent to a partial basement under a house. That means the calculation is already burdened by the fact that the perimeter-to-area ratio of the partial basement (even if it ideally had a square base by itself) is always less favorable than that of a full basement. You can get a basement with half the volume for about two-thirds of the price.
In addition, the garage has the driveway side in front, which practically rules out light wells. Furthermore, vehicles are heavier than furniture, making the "basement ceiling" under a garage more expensive.
Nevertheless, I have also had the thought that it might not be entirely out of the question that the question is worthwhile – and I asked basement builders. There are several prefabricated basement builders who specifically offer mini basements (only a house connection and storage room) and could also manufacture them in the dimensions for under double carports. They told me unanimously that it is not worthwhile. They do sell these things very well where people, like in Stuttgart or Munich, want to save every square meter because of the exorbitant prices for building land and therefore avoid the basement alternative of the above-ground utility room (HAR) – but almost never in places where the pressure to save on floor area is not quite as extreme.
Even a utility room (HAR) behind a carport – that is, not sharing walls with the garage extension, but requiring all enclosing walls on its own – would be cheaper to construct (basically because it is above ground) than such a small basement.
