infratrace.ch/index.php
Joomla 3.9.23

Should show the same content in the module position like desktop view in toolbar-l

Maybe you should first update your Joomla to the latest Version 3.10.8

You can set the mobile logo in astroid-framework-header:

Hello,
you are using the Warp 7 Framework ( no more support) and your logo is on Module position toolbar-l.
But the toolbar is hidden in mobile view. You can remove the _{uk-hidden-small} class from toolbar in the file
_{layouts/theme.php}