Search-based messaging extensions are limited to the UI experience they can provide. One can create multiple search lists in the same extension popup using navigation options horizontally against the top of the display. On the other hand task module based messaging extensions can work with embedded web controls and have full control of the UI.
The new (meaning currently shipping) Stickers app in the Teams client (ability to search for and inject stickers in a conversation) seems to have navigation capabilities not documented? It appears to be a search-based messaging extension due to how it appears in the UI (i.e it is not task module based, as it is not a modal window that appears inside the chat window)
The code available at https://github.com/OfficeDev/microsoft-teams-stickers-app is for the older app and does not implement the current shipping stickers apps UX.
Any pointers on how to get the same left hand side navigation tab experience?