Learn English – Why are days of the week proper nouns

The Oxford Dictionary Online (a proper noun) defines proper noun as

A name used for an individual person, place, or organization, spelled with initial capital letters, e.g., Larry, Mexico, and Boston Red Sox. Often contrasted with common noun.

[It also conflates proper nouns with proper names, and I will not try to sort that our in this answer.]

As the questioner suggests, Wikipedia defines it as

a noun that in its primary application refers to a unique entity, such as London, Jupiter, Sarah, or Microsoft, as distinguished from a common noun, which usually refers to a class of entities (city, planet, person, corporation), or non-unique instances of a certain class (a city, another planet, these persons, our corporation).

I disagree with the criterion of unique. Rather, a proper noun refers to a specific instance of something. During the bestowing of a proper noun, the naming entity seeks to distinguish the individual accorded the proper noun from others in the generic class. When my father named me Dopey, he sought to distinguish me from my brothers, Bashful and Doc, and whoever else might come along later. He didn't especially care that Mrs. McGillicutty, three streets over, had also named her son Dopey. Within our circle, Dopey meant me.

There are very few proper nouns that are truly unique. The number of towns named Springfield, Madison, and Franklin are in the dozens apiece. However jurisdictions resist identical names within their purview. A town near me was named Marion and was called that for decades before someone in the state realized there was another town of the same name and our neighbor was rechristened East Marion.

If it were just me and Dad on a desert island, naming me Boy might work just fine. And maybe if only I were in the room and he said Now listen to me, Son, that would be okay. But when he yelled up the stairs, son, all his kids responded or no one did.

To each of us, Dad means the specific Dad (or sometimes Dads) that relate most closely to us. I had a mother and a stepmother, both of whom I called Mom. I can speak of my two moms, but when I address them or use them without a limiter (such as my mom), I refer to them in the capital.

To sum up, things that are virtually never grouped into a class and have been named to be specific within their own circle are always captialized, even when they are artificially aggregated.

• All the Alberts in the class should stand.
• There are many Washingtons in the US.

These retain their upper case status even when they are converted to a category with a modifier

• The Microsofts of the world will have to rethink their strategies.
• The Roosevelts were a political and social force to be reckoned with.

But common nouns that are used as a term of address or in lieu of a name without modifier are capitalized in those circumstances, but lower case in others. So

• I asked my mother to come over.
• I asked Mother to come over.

It just depends what you mean by "phrase". The everyday sense is an expression (constituent) containing several words, and since the name of an individual person or thing can perfectly well have several words, then yes. A proper noun can be a phrase, with several words. But in such a case, we would be more likely to refer to it as a "proper name" than a "proper noun", because using the word "noun" is often taken to imply that there is just a single word there.

In the grammarians' sense of "phrase", though, a phrase can have just a single word. The subject of a sentence, for instance, is a noun phrase, and of course a sentence subject may be single word. Here, the facts of language impose a certain interpretation for "phrase".

Another instance where the facts of language impose a sense that traditional grammar may find unintuitive is coordinate conjunction. A fundamental rule for how "and" works is that the grammatical category of a coordinate constituent is the same as the category of the two constituents that were conjoined. For instance, two verb phrases conjoined by "and" make up an expression which occurs in the same grammatical contexts as a single verb phrase. This rule works quite generally, and it tells us that the coordination of two nouns, common or proper, is a noun, even though it contains several words.