Hi @dejan85,
I hope you are well today and thank you for your question.
I have tested it on my test site using latest version of Custom Sidebars plugin and default WordPress theme Twenty Fifteen.
I found when using the “Custom Menu” widget the classes “widget widget_nav_menu” are added to the widget container properly.
I think this can be a theme related issue that you are using.
To confirm whether this is theme specific issue you can just try temporary using any default WordPress theme like Twenty Fifteen.
Could you please share me the page URL from your site where you are displaying the “Custom Menu” widget in the custom sidebar so that i can test it?
Kind Regards,
Vinod Dalvi