For anyone reading this, I've figured it out. First, make your properties class inherit from MultiSelectorProperties like so:
public class HiddenMultipleInputProperties : MultiSelectorProperties
Then have your component class inherit from MultiSelectorFormComponent:
public class HiddenMultipleInputComponent : MultiSelectorFormComponent<HiddenMultipleInputProperties>
Follow the prompts in VS to fill in the classes for both. For the view, I made some checkboxes. And now I'm able to compare my custom field component to the native Multiple Choice and (for some reason) Textarea components. Hope this helps someone.