Is "Underground" a proper noun or a common noun?

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.