As semsem said, the simple answer is "no it is not required"... there are ways to get around having an appId associated with the "like" button. Here's my experience working with this. I'm not a Facebook or Open Graph expert, so YMMV.
Why we avoided using the appId on the button:
We're providing an service where we have one website (the engine, as it were) that provides a service distributing online courses to students (customers). Instructors (also customers) who what to use our service to disseminate courses to students can brand the site how they wish, and map their domain to their section of our website that serves those course(s).
As a simplified example: we serve from http://courses.example.com/instructor_name, but we want students to access the content through http://www.instructors-domain.com/. Any courses would be sub-directories off the base URL.
Associating the "like" button with our Facebook App disallows any cross domain shenanigans. While there are valid reasons for doing so, it doesn't work for where we're at in our company and product evolution. So we needed to find a workaround.
We wanted to allow folks to "like" a course, have the "story" point to the appropriate places on the net, as well as get some customization (e.g. "NAME likes an online course on FBAppName"). We basically achieved this. We lost some functionality which we deemed acceptable at this point in our evolution.
The short of it
I used the iframe version of the Facebook "like" button as dictated by the appropriate Facebook developer's page (for the link see semesm's answer for the link, I got no rep). I took their code snippet and manually removed the appId query string in the iframe's src.
In the "liked" page itself (which was the same page that had the "like" button) I used the Open Graph meta tags including specifying the appId. (These tags were specified: fb:app_id, og:type, og:url, og:site_name, og:title, og:description, og:image.)
The og:type was our custom type of the form 'namespace:app_custom_object_name'.
A failed approach
My first attempt was to use what I understand as the preferred method, the "HTML5" tab in the "Get Code" section of the developer's "like-button" page. I tried their method stripping the appId from the appropriate places. This method proved ineffectual.
If the domain doesn't match that in the Facebook App, there will be no "like" button.
If the domain does match, the "like" button will appear. However, it takes 3 clicks to actually "like" something. The first click changes the "thumbs-up Like" icon to a normal anchor with one word that didn't make obvious sense (I forgot what the word was). The second click will brings up the login/authorization window for using our app. The third actually bring up the modern fancy "like" box where you can type in a comment. I didn't find a way around this behavior.
Note that when I specified the appId in this approach on the appropriate domain, it worked as one would expect (though inconsistent with our desired behavior).
I did not try the other two options in the "Get Code" section of the "like-button" page.
Informed speculation and rumor
In my research around this, my overall impression is that requiring an appId is the way of the future for Facebook. Who knows if the old way will be depreciated, probably never, though I didn't find anything in the docs talking about this "legacy" behavior. This makes sense to me with their newer offerings and the advanced tracking that becomes available with this method.
I've seen suggestions that the "likes" used in this manner are akin to second-class citizens... treated as inferior in some respects. In my own experimentation I found the behavior of the fully specified appId (in the "like" button itself) to be different and more accessible and predictable (in terms of Open Graph queries and visibility on my limited Facebook tests) than the partially specified appId. (Again, I've found no solid documentation on this, and did not endeavor to full grok the differences.)
May this info help someone else along. Good luck!