Technically the engine can run without the cli-client being installed. Furthermore, the cli-client can be installed on a system without the engine and use it with a remote engine. The cli-client is nothing than a user interface that interacts with the api endpoints of a docker engine.
Since you are using Windows, the backend for Docker Desktop can either run on Hyper-V or WSL2 (prefered)
You are not supposed to work in the utility vm that runs the Docker Desktop backend. From the user perspective it’s an appliance controlled by the Docker Desktop frontend application.