For standard Joomla content, our automatic insertion of the Like button into articles works exactly how you want, each Like per article in a blog view is for that specific article, not for the whole page.
We don't have the automated version for K2 implemented (yet), but until then, you can achieve that functionality very easily with the {JFBCLike} tags. In each article, at the top or the bottom, simply put:
{JFBCLike href=http://domain.com/link-to-k2-article}
Then, when the Like button is displayed, whether it's in the blog, single-article, or even in a module, the URL that will be Liked, for that article, will be the correct one for the specific article view.
Additionally, we support the OpenGraph tags fully, which let you set the individual image, title, description and other information about a page to Facebook. So, if you did the following in the article:
{JFBCLike href=http://domain.com/link-to-k2-article}
{JFBCGraph image=http://domain.com/best-image-for-specific-k2-article.png}
{JFBCGraph title=Great article about blah}
{JFBCGraph description=Description of article....}
You'd have everything you'd want for a rich post back to Facebook from anywhere on your site. If the user Likes the article from a blog view, since you specify the URL, Facebook should use the Open Graph information from that specific article, and you'll be set.
Hope this all helps, and yes, we've got you covered