How to add tabs to panels setting page? I want to add additional vertical tabs (marked with red) in screenshot, but have had no luck with google or documentation.


How does one add tabs here?
Those vertical items are internally called "operations" and are implemented by task_handlers plugins although some of them are hardcoded in page_manager. You can see some examples in:

ctools/page_manager/plugins/task_handlers/http_response.inc 
panels/plugins/task_handlers/panel_context.inc


You may also find more info in page_manager_render_operations()  docstrings.