Three men sharing a meal should have a cup.
When one eats alone he doesn’t make a Kiddush but three or more should make the higher connection with the Kiddush.
The Zohar explains that Binah is the third of the ten Sefirot counting from top to bottom. Because of that, less than three do not require a cup.
“וְקָרָא זֶה אֶל-זֶה וְאָמַר, קָדוֹשׁ קָדוֹשׁ קָדוֹשׁ יְהוָה צְבָאוֹת; מְלֹא כָל-הָאָרֶץ, כְּבוֹדוֹ”
“And one called unto another, and said: Holy, holy, holy, is the LORD of hosts; the whole earth is full of His glory”
‘Holiness’ is expressed in three as we do in the prayers of Kedusha that we recite daily and Keter on Shabbats and Holidays.
The Torah requires three types of people to read it in holiness, כהן Kohen, לוי Levi and ישראל Israel. Their initials make the word כלי, meaning vessel and they also represent Right, Left and Center columns of the Tree of Life. The Bible has three parts, Torah, Prophets and writings. The Torah was given on the third month, Sivan (Gemini).
A cup of blessing is the aspect of the last H ה of the YHVH name. The quantity of wine should be no less thatn רביעית לוג, ‘fourth log’. Log is the volume of 4x4x4 fingers. There is no exact measurement and the average of the common known estimation is about 100cc.
When 10 or more people share a meal they make the blessing of the food adding “נְבָרֵךְ לֵאלֹהֵינוּ” “we shall bless our God” at the beginning. The explanation is that with ten we express the ten letters of the name of God יוד-הא-ואו-הא. When we do that then we need to add the name ‘אלֹהֵינוּ’, ‘Our God’.