You are not using a DisplayLink enabled dock or adapter
USB C docks may not all use DisplayLink technology and instead rely on USB "Alternate mode" for a video output. Alternate mode is not supported by all phone eg the Google Pixel phones up to the pixel 3. Unless you have a DisplayLink enabled dock, then it will not work with the DisplayLink app.
Look for the DisplayLink "Plug and Display" logo on your product to ensure it is DisplayLink enabled.
The phone is not a USB host
Usually phones are connected to a computer eg to transfer files. In this use case, the computer is the USB host and the phone is the USB device. When connecting to a DisplayLink enabled dock, the phone needs to become the USB host and the dock or adapter is the USB device.
To switch the phone into a USB host, then a USB OTG (On The Go) adapter is used. With USB C this should happen automatically, but is not supported by all phones. This is covered in more detail in this article.
The OnePlus phones also do not automatically switch into OTG mode, which will prevent a DisplayLink dock working automatically with these phones. There is an article about how to enable OTG on the OnePlus series of phones so they can be used with DisplayLink docks.
The Phone cannot deliver enough power to a DisplayLink enabled device
This is particularly relevant to USB bus powered monitors using DisplayLink, but can apply for any DisplayLink enabled product which is bus powered.
Android phones limit the amount of power delivered through the USB port, to save battery power. This means that DisplayLink products which are USB bus powered will not properly power up and be seen by the phone. To work around this, USB bus powered products should be connected to an externally powered USB hub, then connected to the phone.