Hello,
the login module is mapped to a Responsivizer template position so obviously it shows up.
Unpublish the login module or the exclusions button to remove it from the mobile template.
Looking a bit deeper at the extension you will see that it works using modules matching between desktop/mobile template so you can build the mobile version of the website including only required elements.
An overview video tutorial to get basic concepts can be found here:
https://youtu.be/B1fqydcZYlI
Best regards
John