Shulchan Aruch (OC 65) says to say "Sh'ma" with the congregation when they say it, and Mishna B'rura 9 there says the same is true of other things recited as a congregation such as "Ashre" and "Alenu". (Indeed, it is common practice that one recites "Alenu" when the congregation does even if he is not at that point in the service or is not praying at all.) In what sense are the latter two considered prayers that are recited communally (more so than most of the rest of the prayer service), that the Mishna B'rura mentions them as communally-recited prayers? That is, what makes them communal, whereas other prayers aren't necessarily so considered?
Regarding Ashrei see this M.Y. answer. He cites Ramba"m and, more importantly Talmud Brachot which states that the beginning words refer "to those who dwell in your house". This is a reference to those that come to the Bet Midrash and to shuls. The Gemarah states that people would spend time meditating in the shul prior to the start of davening. Ramba"m clarifies the requirements moreso stating rules about if the Chazan stands or sits and what the congregation does. So, this definitely seems like a communal prayer.
Aleinu has an interesting history. It seems that it began as an individual prayer. This article indicates that the first mention of Alenu appears in Siddur Rash"i siman 419. Originally, Psalm 83 followed the Chazzan's Kaddish (Kadish Titkabel) and *aleinu was considered an "extra" tefillah recited by the individual, similar to what is commonly done now regarding the 6 Remembrances, 13 principles of Ramba"m, etc.
This article indicates that the communal concept of Aleinu began during the Crusade period. The main purpose was to implant the concept of unity of G-d in the people's minds esp. during the time that the Crusaders where trying to deter our beliefs. Because of this concept of unity of G-d, many connected the requirement to say Aleinu only during those tefillot that had Shema, as they saw a relationship of the two. Thus, since Mincha had no Shema, they did not recite Aleinu at the end of Mincha.
Note that Eliyahu Rabba 237:4 says that even if one has said Aleinu individually, one who joins a congregation later on, must say Aleinu with the congregation.
So, it seems that we are following the latest custom, and it would seem that reciting Aleinu is definitely "communal".