Everything in WP has an ID. As long as your theme is using body_class you probably already have built-in classes that will cover your scenario.
For example:
- Pages will include a
page-id-##class, where ## is the id specific to that particular page. - The homepage will include a
homeclass and if it’s your main blog archive also ablogclass. - Posts and custom post types will include a
postid-##class, where ## is the id specific to that particular post/CPT. - Categories will include a
category-##class, where ## is the id specific to that particular post.
So, you should be able to use built-in functionality to identify content. All your theme has to do is, in header.php, use body_class like this:
<body <?php body_class(); ?>>