Names are supposed to be proper nouns because they refer to a unique entity, right? But what about when the condition of specificity is not applicable? Take the word "Albert". It's supposed to be a proper noun. Why? There are no doubt thousands of people with that name in the world. If you know more than one Albert then just the word in not sufficient for you to know who precisely is being referred to. You'd naturally want to know the full name or any other unique identifying characteristic to pin down the person concerned. For that matter, even the full name (say "Albert West") is something the person is likely to have in common with others on this planet of seven-odd billion people. So while "Albert" (or any other name for that matter) might be a noun, why is it considered a proper noun?
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.