When we say in the shema, "ה' אחד", we mean that God is one in the fact that he is indivisible in such a way that is unimaginable. (See Mishna Torah, Hil. Yesodei Hatorah 1:7.)
Here, the intention is not the oneness of God as an entity, but rather God's uniqueness in that he is the only "godly" being and that no other "gods" have such power. So the verse "ביום ההוא יהיה ה' אחד ושמו אחד", means that eventually, God and his name will be one, i.e. He will be recognized as the only God and will be served by all the nations as such. (See Abarbanel, Zecharia 14:9.)
Why do we pick this verse to say at the end of Alenu and not others? Well, we are trying to use this verse as a source for the fact that not only will all the nations eventually serve God as their king(ויקבלו כלם את עול מלכותיך), but also that all other gods will be worthless and destroyed and forgotten (לְהַעֲבִיר גִלּוּלִים מִן הָאָרֶץ וְהָאֱלִילִים כָרוֹת יִכָרֵתוּן). The only other p'sukim I can think of that relate the nations' eventual serving of God (Tzefania 3:9; Yeshayahu 2:2) don't really specify that God will be unique in their serving him; they don't specify that the nations won't still be serving their old gods as well. (Ch. 2 in Yeshayahu does bring this out, but it takes several p'sukim to get the message across, and it also doesn't emphasize God's kingship, unlike in Zecharia where it is all nice and compact in this one pasuk.)
This verse: "ביום ההוא יהיה ה' אחד ושמו אחד", as it was explained above, brings across that very message that God will be served by all the nations as the one true God.
[Note that the gemara (Pesachim 50a) asks your first question about God's oneness. The answer it gives (על דרך דרש) is that today God's name is spelled "ykvk" and pronounced "adonai". In the Messianic era, His name will be pronounced the way it is spelled.]