Continuous Music throughout Site Browsing

Is it possible to have the widget continuously play a song even if the user changes a page on the site?

no, this is not possible; however, if you want to autoplay the music as soon as the page is loaded you can add the “autoplay Attribute” to the audio tag.

if you look at the source code, you can see the audio tag used:
<?php echo '<audio preload="" controls="" src="'.$url.'"><p>Your browser does not support the audio element.</p></audio>'; ?>
so simply add autoplay after control, so it would be like :
<?php echo '<audio preload="" controls="" autoplay src="'.$url.'"><p>Your browser does not support the audio element.</p></audio>'; ?>
this will autoplay the song automatically, hope that helps