That’s not exactly what the theme customizer is for. The theme customizer should be reserved for big (global) items. Like the background color of every page, not just a single page. There isn’t a way (that I know of) to change things on a per page basis.
For things that are page specific to the page/post, I’d recommend using the custom post meta. Here’s a pretty good example: http://wp.smashingmagazine.com/2011/10/04/create-custom-post-meta-boxes-wordpress/