Question: When a client edits a featured image, they notice that they have the ability to set its "Alignment". If they were working with standard images embedded via the Editor, then this wouldn't be a problem. But since it is a featured image, WordPress doesn't pass in the appropriate class.

Is there a way to know which alignment option was set? If so, I can use that to add a class to the featured thumbnail accordingly.

Answer: 
Short answer: **you don't**.

The alignment settings only apply to images that the user *adds to the Post Content*. The Featured Image is *incorporated into the template* and is *not added by the user* to the Post Content. 

The Featured Image is directly integrated into the template, usually as a "post thumbnail" displayed on archive index pages, or as a "post header image" displayed above the post on archive index pages and/or single blog posts. In such cases, the Theme developer accounts for the *presentation* of the Featured Image via the template markup and CSS.

(That's not to say that the user can't manually add to the post the image used as the Featured Image; the user certainly *can* - and in which case, the manually added image will use/apply the alignment setting.)
I am sort of confused with your question but I hope that this link will help.

<http://codex.wordpress.org/CSS#WordPress_Generated_Classes>