Related nodes by taxonomy term in a view block of drupal7
There will be a significant amount of customization (in the form of a module) to get this done through views. I've created a sandbox module that you can install to get things working and so you can play around with it (see views.api.php in the views module).

Once enabled you should find a new contextual filter value called "The node whose taxonomies you want to relate". This contextual filter default will take the current node ID and return nids of nodes that share the same terms in vocabularies that you select. Therefore, please set the view display as a block.

If you need to know how to group by term using views, please see this answer.