To work with mobile templates, you need to install the Mobile and Mobile Adaptive extensions. They have a single package for all versions of Joomla. But the templates (Mobile Basic, Mobile Elegance, etc.) are different for different versions of Joomla. I hope you have downloaded and installed the correct extensions: Mobile 3.3, Mobile Adaptive 3.3, and Mobile Basic 3.2 for Joomla 4x.
To activate a mobile template, open the Adaptive tab in the Mobile settings and select the template you want to use in the mobile version. Don't forget to save your settings.
When installing Mobile Adaptive, a menu module (mainmenu) is automatically created in the mj_panel position. You can either change the displayed menu in the settings of this module instance, or, if necessary, place other menus in other positions of the template.
You can check if everything works correctly with the Mobile Basic template, then install Mobile Elegance and select it in the Adaptive tab.