Custom backgrounds for sections

I started recently with building WP plugins. After two simple first plugins it’s now time for something more challenging. I’m building a “one-page-layout” plugin.
In order to accomplish this I have created custom post types which I call ‘Sections’. They can be implemtented either by a shortcode or by a template tag/function.
They can be created an edited in the WP backend, just like pages and posts and they are available in custom menu’s.
Everything works, so far so good.

Now my question
In a one-page-layout, every section often has it’s own background. So I would like my plugin to have the option to set a custom background for every single ‘section’. Just like the default WordPress custom background option, but this time on “(custom) post level” instead of website level.

Of course I could write all the neccesary code from scratch, but I rather use eventual WordPress functions for this, so that I don’t need to re-invent the wheel.
So, is there a way to implement the standard WP custom background option in my custom posts? And if it is more complicated than that, where can I find functions or documentation on how to do this? What are the keywords to look for?