Julie here from Shopify Support.
It's definitely possible have a music player continue to play music on all of your pages and the easiest way to implement this is through an app. There are actually a few music player apps in the Shopify App Store. It looks like you already have a music player on your site — did you use an app to add it? I'm curious to know which one!
I took a look to see which media player apps are able to continuously play music on all pages, and I managed to find a few options for you:
All three of these apps have great reviews and are easy to install. They offer customization options so you can make the music player match the look and feel of your brand. You'll also be able to adjust the location of the player and choose between various autoplay settings. To add music, you'll just need to upload files, since they don't offer streaming options.
It looks like you're a radio station, so if you're wanting to broadcast your station live, then I suggest checking out Elfsight Radio Player. Although it's not quite clear whether this app will give you the option of continuously playing on each page, it might be worth finding out! You can always install it and take advantage of its 7-day free trial, or you could reach out to the developers directly. Keep in mind that if you install the app, you'll need to uninstall it prior to the trial expiring if you don't want to be charged their monthly subscription fee.
I hope this provides you with a few options worth considering! Let me know if you have any further questions.