Monday is considered to be a proper noun (in English the capitalization is a dead giveaway):
A proper noun or proper name is a noun representing a unique entity (such as London, Jupiter, John Hunter, or Toyota), as distinguished from a common noun, which represents a class of entities (or nonunique instance[s] of that class)—for example, city, planet, person or corporation)...
Which nouns are considered proper names depends on language. For example, names of days and months are considered proper names in English, but not in Spanish, French, Swedish or Finnish, where they are not capitalized.
Monday is a unique entity (representing the second day of the week), whereas the corresponding common noun, day, represents any day of the week.
In conventional resources on common and proper nouns, they are treated as two separate subcategories of noun.
Some sources do consider converting one kind of noun to another. Here is ThoughtCo author Richard Nordquist on going from a common to proper noun. The only applicable verb in the article is capitalize, referring to the change in form that signals a proper noun:
Many great authors have used the idea of capitalizing common nouns and making them proper to characterize specific inanimate objects or take a concept like "Great Places" and make them into a physical place in a fictional world.
However, in more academic contexts linguists have referred to the creation of proper nouns (single words) or proper names (noun phrases) as naming. Adrienne Lehrer in "Names and Naming: Why We Need Fields and Frames" describes the conventions of naming:
A common basic distinction in noun subclasses is between proper and common nouns. Often little more is said about the difference, as in contemporary generative grammar, where the difference is characterized by the feature +/- Proper. However, if we look at a wide range of names and at the processes for naming, we discover that the difference between common and proper nouns is anything but clear-cut; and moreover, the vocabulary is not neatly divided.
Another author, Willy Van Langendonck, describes this process as naming when summarizing a theory of proper name formation in Theory and Typology of Proper Names:
Thus, the difference between proper nouns and common nouns is basically a difference of dénomination (naming), i.e. proper names should be defined in terms of the (instructional) meaning X called Y, but not common nouns.
So naming forms a proper name or noun from (usually) a common noun or (occasionally) other parts of speech.
The reverse process is appellatization, described in The Dictionary of Historical and Comparative Linguistics as
The conversion of a proper name into a common noun
and cites aspirin and Kleenex as examples. Its verb form (appellativize) is turgid and technical.
Best Answer
The definition does not say that the word "underground" is specific to London.
"The Underground" in London is a proper noun (that's why it's capitalized).
"The Metro" in Paris is a proper noun (again, that's why it's capitalized).
If you're talking about underground train systems in general, what word do you use in England? You can't use "subway" because it means something different. So you use "underground" or "metro" — they're both common nouns in this usage.
If you ask whether anybody actually uses "underground" for anything other than the system in London, it seems to be incredibly rare. But the dictionary definition clearly suggests that this is one possible usage, and it would be this usage that is a common noun rather than a proper noun.