The USB-C charging cable is not included. You need it if the USB power output of the connected computer is too low for the connected USB devices. This can be a topic for laptops in particular.

No, just the following: 1x Ugreen CM662 Switch, 1x remote control, 2x USB-A cable

It does not come with a usb C cable. But it's not necessary. I can even use it to switch between a docking station and DVD burner.

Look at the pictures in the article, there's a "What's in the Box" included. It is important that you connect it to the USB port that supplies the most power, so you won't need a charging cable.