I. The Issue
It is common in Israel for many people to greet each other on Purim and Chanukah (and Israel Independence Day) with "Chag Samei'ach" (Happy Holiday). There is a reason to oppose this practice, based on a halakhah regarding Shemini Atzeres. I'll try to explain the reason for the objection and why I think it doesn't apply.
II. The Objection
The Shulchan Arukh (Orach Chaim 668:1) writes that in the prayers we refer to Shemini Atzeres as "Yom Shemini Chag Ha-Atzeres". The Rema writes that the Ashkenazic practice is different, to say "Yom Shemini Atzeres Ha-Chag". The reason he gives for this is that the other way implies that Shemini Atzeres is a "chag" in the technical sense of having a chagigah sacrifice, which is not correct. Therefore, rather than call the day a "chag", we should refer to it as the end (atzeres) of Sukkos, which is a "chag". (The Vilna Gaon [Ma'aseh Rav, no. 268] follows the vast majority of medieval authorities who support the first version and not the Rema's.)
However, the Pri Megadim (ad loc., Mishbetzos Zahav 1) disagrees with the Rema and says that both ways can be understood as implying that Shemini Atzeres is the end of Sukkos, which is a "chag". And, indeed, the Maharil explicitly adopts the Shulchan Arukh's way of saying it but explains it as the Pri Megadim does. However, the Pri Megadim adds that we should keep the phrase "Shemini Atzeres" connected and uninterrupted, like it is in the Torah. (He also says that we should say "Atzeres" and not "Ha-Atzeres", but I have never seen a prayerbook that has it that way.)
According to the Rema and Pri Megadim (and many others), we have to be careful to not refer to Shemini Atzeres as a "chag" because it lacks a chagigah sacrifice. If that is the case, then we should also make sure not to call Purim (and other days) a "chag" because it also doesn't have a chagigah sacrifice. Hence the objection to the greeting "chag samei'ach" on days that are not technically chag.
The Bekhor Shor (Sukkah 46b) and Birkei Yosef (Orach Chaim 668:1) bring many proofs that the Talmud understood Shemini Atzeres as being considered a "chag" despite its not having a chagigah sacrifice. If they are correct, then the Rema's objection falls away (although the Pri Megadim's remains). However, this does not necessarily mean that the greeting is appropriate. Just because the Torah called Shemini Atzeres a "chag" doesn't mean that any day can be called one; it is, after all, still a full-fledged holiday.
III. The Counterargument
Despite all this, I don't think that this objection is convincing. The Shulchan Arukh (Yoreh Deah 220:20) rules, based on a Yerushalmi, that someone who vows not to have wine on "chag" is not allowed to drink it even on Shemini Atzeres. The Taz (ad loc. 26) writes that this is not relevant to our discussion of whether Shemini Atzeres because vows follow colloquial language (leshon benei adam). Even if Shemini Atzeres is not technically considered a "chag", if people regularly call it one then in a vow it is considered a "chag". The language of prayer, however, is different and cannot follow colloquial usage.
In addition to the obvious, what we see from this Taz is that there is no objection to calling Shemini Atzeres a "chag" even if it is not technically one, as long as you don't do it in prayer (the Taz certainly disagrees with the Bekhor Shor quoted above). If so, there should also be no objection to calling Purim (and other days) a "chag", by saying "chag samei'ach", if that is the colloquial usage. Like with vows and Shemini Atzeres, it should not matter whether or not a chagigah sacrifice is brought on that day. It is all a matter of common usage.
