Although you already found your solution, I figure I might as well post the solution for this plugin here. This happened to me as well, and I found out it was actually the result of my theme not creating widgets with proper class identifiers and the plugin depending on the theme in classing its widgets (not explicitly including aside/div class tags on its own).
An easy fix is to go to the social-media-icons-widget/lib/widget.php and changing the opening_tag and closing_tag lines to this:
<?php echo apply_filters('social_icon_opening_tag', '<aside class="social-icons-widget"><ul class="'.$ul_class.'">'); ?>
<?php echo apply_filters('social_icon_closing_tag', '</ul></aside>'); ?>
