How do you apply multiple skins to a site via the “templates” selector using stylesheet switching?

Sure it’s possible. The quick way would be to add a custom field to the postmeta (in the “Custom Fields” area of the Edit Post/Edit Page page) and in your theme get the value using get_post_meta().

The slightly-less-quick way would be to add a wrapper to this Custom Field value by putting your theme selector in a post meta box.