Is the observance of the eighth day of Chanuka related to the way in which outside of Israel festivals are observed for two days?
This is an interesting suggestion. But although the day is certainly a special day, for reasons I’ll write below, it doesn’t seem to be for the reason you suggest. I say this because even in Israel, Chanuka is observed for eight days. Also, some commentaries actually go so far as to explain that Chanuka is not observed for an additional day outside of Israel because its holiness is derived from the merit of the Jews’ own deeds, and therefore, even in the Diaspora, its holiness is equally experienced and shared by all.
So what is special about this day? It is even referred to by a special name, “Zot Chanuka” [literally, “This is Chanuka”] because the Torah portion read on this day concludes with the phrase, “zot chanukat ha’mizbeach” which means, “this is the dedication of the altar”.
The Sages interpreted this verse allegorically: “This is Chanuka” means that this, the eighth day, is the essence of Chanuka. The number seven alludes to that which is physical or temporal, like the seven colors of the spectrum, the seven musical notes, and the seven days of the week. Eight, however, alludes to that which transcends nature, such as the covenant of the brit performed on the eight day. So the eighth day of Chanuka epitomizes the transcendence of spirituality over physicality, right over might, and sanctity over impurity.
That being said, we do find an idea which is similar to your suggestion. Namely that the last day of Chanuka, if not like an additional holiday of the Diaspora, parallels Shemini Atzeret which follows the seven days of the festival of Sukkot. Just as Shemini Atzeret contains the essence of all the festivals that precede it – the atonement aspect of the Days of Awe and the joyous aspect of Sukkot – so too does this last day of Chanuka contain all of the aspects of happiness, salvation, praise, and thanksgiving of the previous days.