One issue unique to Yom Kippur which falls on Shabbos is the question of whether a seriously ill person who must eat should recite kiddush. Most poskim opine that on a regular Yom Kippur one who is forced to east should not recite kiddush – kiddush recited over a cup of wine is a Rabbinic takkanah, and the Rabbis never instituted kiddush when most people are fasting. However, if Yom Kippur falls on Shabbos, R’ Akiva Eiger suggests that kiddush perhaps should be recited. While there was never a takanah of kiddush on Yom Kippur, there is a takanah to make kiddush every Shabbos, and perhaps that takanah is in force across the board on any day that has kedushas Shabbos.
The way I have heard this issue explained is that R’ Akiva Eiger seems to view the kedushas hayom of Shabbos as completely separate from that of Yom Kippur - the fact that the days coincide has no impact on their respective laws. One could make the counter-argument that when two days with different kedushos coincide the kedushos merge – there are not two separate coincidental kedushos, but a synthesis kedusha of Shabbos-Y’K, different from every Shabbos of the year and different from a regular Y”K. (Parenthetically, and a different discussion, the Rambam holds that Shabbos and Yom Tov are two separate kedushos but Shabbos and Y”K is kedusha achas, but I don’t want to get into the nafka minos of that now.) The fact that kiddush is recited when there is kedushas Shabbos is not a proof that kiddush should be recited when there is a synthesis kedusha, Shabbos-Y”K.