We all know the Beis Yosef's question about why we celebrate eight days, when the miraculous long-burning oil only lasted seven days more than it normally would. The Aruch Hashulchan addresses this question in OC 670:5. He brings from the Sefer Chashmona'i that in the year before Matisyahu's rebellion, Antiochus had prevented the korban celebration of Sukkos and Shemini Atzeres. Therefore, when the Jews were able to re-inaugurate the Beis Hamikdash, they intentionally celebrated for eight days, in order to show that they were making up for the lost days of Sukkos and Shemini Atzeres. The eight days of burning, then, were simply a heavenly ratification of their decision to establish this eight day holiday. (This is actually made clear in the Megillas Chashmona'im.)
With this we can finally understand what Shammai means by saying that we should start with eight candles and go down one every day just as was done with the bulls that are brought on Sukkos. Everyone reading the Gemara is puzzled by this association, because this reverse progression is unique to Sukkos, and why would Chanuka davka reflect the singular rules of Sukkos? But now we understand that Chanuka was viewed as a stand-in, as a commemoration, of the Sukkos holiday and korbanos that they had been prevented from bringing.
This also explains why we find dinim of hiddur on Chanuka that we generally don't find in other dinim. The reason is, again, because Chanuka is, to some extent, a quasi-Sukkos, and Sukkos is a holiday when hiddur on the esrog and all the minim is stressed to an unusual extent.
Micha Berger of http://www.aishdas.org/asp/2006/11/chayei-sarah-kibbush-and-chizuq.shtml commented that he heard this association from Rav Aharon Soloveichik in 1993, and expanded on it with something else he heard from him. What follows is from Micha's website, Aspaqlaria, at the above link.