Is Home screen 3D Touch menu on supported iPhones user customizable? Attached is a screenshot of what appears when I force touch (3D Touch) on the 2Do app running on an iPhone X. Annoyingly, the action view doesn’t contain Search item. Can I alter the menu so that it has a search option (like Apple Music - screenshot attached)
There is no setting in iOS to customize the action view displayed when 3D touching an app icon on iPhone Home screen. It is up to the app what items are displayed. For 3rd party apps, iOS generally displays a Share app name item.

However, an app may provide settings within itself to customize what items it displays. Thus, it's best to look within the app to search for any relevant setting.

2Do app doesn't appear to offer any such setting.