Why are the 13 attributes of mercy only said with a Minyan?
The Rashba (1235 – 1310; Barcelona) was asked whether one can say the thirteen attributes when praying alone. He answered that as long as one says it in a manner of praying and pleading it should not be said when praying alone, as with all Devarim She’Bikdusha (a form of sanctification) which require a Minyan (quorum of ten men). But if he wants to say it in a manner of general reading he may say it as we say also the Ophanim (wheels) and verses of sanctity in a manner of reading. The source that the saying of the thirteen attributes of mercy is considered a form of sanctification is in the Talmud Tractate Rosh Hashanah 17b: And the Lord passed by before him and proclaimed (Exodus 34, 6) R. Yochanan said: Were it not written in the text, it would be impossible for us to say such a thing; this verse teaches us that the Holy One, blessed be He, drew his robe round Him like the reader of a congregation and showed Moses the order of prayer. He said to him: Whenever Israel sin, let them carry out this service before Me, and I will forgive them. The source that Devarim She’Bikdusha are said only with a Minyan of ten is in the Talmud Tractate Berachot 21b and Megilah 23b: Whence do we know that a man praying by himself does not say the Sanctification? Because it says: I will be hallowed among the Children of Israel (Leviticus 22, 32); for any manifestation of sanctification not less than ten are required. How is this derived? Rabinai the brother of R. Hiyya b. Abba taught: We draw an analogy between two occurrences of the word ‘among’. It is written here, I will be hallowed among the children of Israel, and it is written elsewhere. Separate yourselves from among this congregation (Numbers 16, 21). Just as in that case ten are implied, so here ten are implied. And this is ruled in Halachah Shulchan Aruch Orach Chaim 55, 1 and in Mishnah Berurah 2. And so is ruled in Shulchan Aruch (Orach Chaim 565, 5) that an individual is not allowed to say the thirteen attributes in prayer or in request for mercy because they are a form of sanctification [which may not be said by less than ten men – Mishnah Berurah ibid 11]. On the other hand, if one wants to say them merely as reading he may say them [I.e. tunefully and with the traditional musical notes – Mishnah Berurah ibid 12]. (שו"ת הרשב"א ח"א סימן ריא, רב נתן גאון בסדר רב עמרם ח"ב סי' נט, ובטור וב"י או"ח סימן תקסה סעיף ה, ובדרכי משה סק"ה ציין שכ"כ בתרומת הדשן סימן ח).