The word "class" is being used in two different senses here.
Noble, as a social class stratum in a Feudal/Medieval setting, is a different use of the term than "Class" for a PC such as Druid, Fighter, Ranger, Wizard, etc, that has a progression from level 1 - 20 with game mechanical benefits.
Answer: Noble is a background for D&D 5e, not a PC class.
Any PC can come from a Noble social class if the Noble Background (Chapter 4) is chosen.
When you are dealing with a fantastical setting that has kings, queens, princes, lords, and such broadly rooted in the typical feudal / medieval / Renaissance cultural setting from Western Civ, class differences are embedded in the default setting. As you can see in the text for the Noble background:
You might be a pampered aristocrat unfamiliar with work or discomfort, a former merchant just elevated to the nobility, or a disinherited scoundrel with a disproportionate sense of entitlement. (Basic Rules, p. 42)
Aristocrats and nobles come from a particular social class in the default setting. Commoners come from a different layer of the societal strata. See also this feature:
Feature: Position of Privilege
Thanks to your noble birth, people are inclined to think the best of
you. You are welcome in high society, and people assume you have the
right to be wherever you are. The common folk make every effort to
accommodate you and avoid your displeasure, and other people of high
birth treat you as a member of the same social sphere. You can secure
an audience with a local noble if you need to.
Aside: Commoners have an NPC default description, as do Nobles. That is in the Monster Manual (or the Monster / NPC appendix, Basic Rules) and is separate from the character creation of a PC.