The Shulhan Aruch (Orah Haim 208) addresses the case of somebody who drank wine, such that he is required to recite Me’en Shalosh – specifically, the Beracha of "Al Ha’gefen" – and mistakenly recited Birkat Ha’mazon instead of Me’en Shalosh. Surprisingly, the Shulhan Aruch writes that the person has fulfilled his obligation through his recitation of Birkat Ha’mazon, and he does not then need to recite Me’en Shalosh. The reason, the Shulhan Aruch explains, is because wine has a nourishing quality, and Birkat Ha’mazon thanks G-d for nourishing all His creatures ("Ha’zan Et Ha’olam"). Since the theme of Birkat Ha’mazon is relevant to wine, one who mistakenly recited Birkat Ha’mazon instead of "Al Ha’gefen" has fulfilled his obligation.
The Shulhan Aruch applies this as well to the case of one who mistakenly recited Birkat Ha’mazon after eating dates, instead of reciting the Me’en Shalosh required over fruits of the special species ("Al Ha’etz Ve’al Peri Ha’etz"). The Gemara states that dates have nourishing properties, and thus Birkat Ha’mazon fulfills one’s obligation after eating dates, just as it does after drinking wine, even though one was to have recited Me’en Shalosh.
Startingly, the Shulhan Aruch does not apply this Halacha to the case where it would seem most intuitive – where one mistakenly recited Birkat Ha’mazon instead of "Al Ha’mihya" over a "Mezonot" food. Products made from flour most closely resemble bread, are nourishing, and are even referred to as "Mezonot," a word related to the verb "Z.N.," which denotes nourishment. And yet, specifically in regard to "Mezonot" foods, the Shulhan Aruch writes that one who mistakenly recited Birkat Ha’mazon instead of Me’en Shalosh has not fulfilled his obligation, and must then recite "Al Ha’mihya."
The Be’er Ha’gola commentary to the Shulhan Aruch (by Rav Moshe Rivkes, d. 1671) writes that the Shulhan Aruch’s ruling is a "Peleh" ("wonder"), and disputes this position, arguing that it is impossible that Birkat Ha’mazon suffices for wine and dates, but not for "Mezonot" foods. This is also the conclusion of the Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933), who offers an explanation for the Shulhan Aruch’s position, but nevertheless rules that one who recited Birkat Ha’mazon instead of "Al Ha’mihya" has nevertheless fulfilled his obligation. This is also the ruling of Hacham Ovadia Yosef, who emphasizes that this applies both to baked "Mezonot" products, such as cookies and cake, and to cooked products, such as noodles. It applies to fried donuts, as well (listen to audio recording for precise citation).
Summary: Generally speaking, one who ate food other than bread and mistakenly recited Birkat Ha’mazon instead of a Beracha Aharona has not fulfilled his obligation, and must recite the proper Beracha Aharona. The exceptions to this rule are wine, dates, and all "Mezonot" foods. One who mistakenly recited Birkat Ha’mazon instead of Me’en Shalosh after eating dates or a "Mezonot" food, or after drinking wine, has fulfilled his obligation and does not then recite Me’en Shalosh.
