Going to the properties of VDI machine, Client Assignment, selecting user for permanent assignment shows a dialog box where Computers/Client Names is grayed out.
The reason that option is grayed out because in properties of the Desktop Group, that VDI machine belongs to, there's a setting to assign by Users. Please change VDI group Client Assignment method from user to Device Name. Going back to the properties of VDI machine, Client Assignment should show "Select a device name..." instead of "Select a user ..." and option will be enabled.
Important: changing the target for Client Assignment of the Desktop Group will remove all persistent assignments in that group because of different target type. Warning dialog will be provided.