Hammerklavier Sonata is a whole world itself. It's a tremendous piece, which requires from the pianist the best of him/her, both phisically and mentally. Regarding to your question, I think the key word hear is fugato.
A fugato is a passage written in fugal style whithin another work that is not a fugue, acording to Wikipedia. So how do we know when a passage is a fugato and not a fugue itself? Well, keep listening. A canonical fugue (as Bach's fuges from Well-Tempered Klavier, Art of Fuge, etc...) has some fundamental parts that give the piece the structure of a fugue. Exposition, episode, development...all of this is needed for a fugue to be considered as one.
Now, let's go back again to Beethoven's little passage on the op. 106 Sonata. As you said, judging it only for its length (no more than five bars), there's little to say about it. It's a contrapuntual passage, two voices, that seem actually independent one from the other, but that's very far away from becoming a fugue. There isn't a neat exposition of the voices, oposed to what happens when the real fugue is developed a few bars later, where each voice is clearly independent and developes throughout the music. If I had to name that passage, I would say it is closer to the subject of a Bach's Invention than a fugue.
So, why would Beethoven compose something like that? To be honest, I don't know. Maybe he needed a motive to finish with the dramatic atmosphere he created during the third movement, and if that was his porpose, he did it quite well. Or maybe he was just trying to develope a subject for a fugue, but he was not convinced at all, but instead of moving to another subject he decided to leave it there as a premonitorious passage of what it would come later.
And to answer your final question, I have to say that I couldn't find in any other composer such a short passage composed in fugal style. Beethoven's op. 111 Sonata has some parts in the first movement where you can see contrapunctual style as well. An on the other hand, the Gross Fugue, the first movement of his op. 131 string quartet, or the last movement of his 9th Symphonie are fugues respectively, but none of then as short as that little passage on the Hammerklavier Sonata. The closest I've found, is
from Liszt's Piano Sonata in B minor. I think it's the best example of what a fugato is. Here is an audio from Daniil Trifonov playing it where the fugato passage starts. Though it's not my favourite version, I hope it helps.
