Hi Jeremy,
Greetings from Syncfusion support.
We have implemented the Popup as a separate plugin which is act as a dependent component for many Syncfusion components such as DatePicker, Dialog, MultiSelect.. etc. We can customize the popup UI based on our application needs. So, please share the details for how you want to customize the UI of the popup in the MultiSelect component. Also, please share the use case for create the external popup component instead already implemented UI that will help us to proceed further for the requested requirement.
Also, please find the documentation and demo link for DropDownTree component which have same UI for the requested requirement.
Regards,
Berly B.C