No, but you still keep that bonus for Wild Shape.
Each class has a section called Features, that lists all its class features. These include, for a Barbarian, Rage, Unarmored Defense, Reckless Attack, Danger Sense, etc. The Proficiency Bonus is not listed there.
Chapter 7 of the PHB describes it as
Proficiency Bonus: Characters have a proficiency bonus determined by
level, as detailed in chapter 1.
The character advancement table in PHB, pg16, shows your Proficiency Bonus regardless of class.
When your character gains a level, his or her class often grants additional features, as detailed in the class description. In addition, every character’s proficiency bonus increases at certain levels
The Character Advancement table summarizes the XP you need to advance in levels from level 1 through level 20, and the proficiency bonus for a character of that level.
The Multi-Class chapter also lists Proficiency Bonus based on character level, not class levels.
Your proficiency bonus is always based on your total
character level, as shown in the Character Advancement
table in chapter 1, not your level in a particular class.
It also does not list Proficiency Bonus under special rules for Class Features gained, as Marq points out.
Just for completeness's sake, monsters have Prof Bonuses too, although based on their CR and not Character Level. It is shown in the Monster Manual, pg.8, and on the DMG, pg. 274 (thanks to Slagmoth for the tip).
Keeping the proficiency bonus for Wild Shape
Despite not being a Class Feature, it still carries over on Wild Shape or Shapechange, as confirmed by lead rules designer Jeremy Crawford on Twitter. The ability/spell mentions you take your original proficiencies, and that includes this Proficiency Bonus.
While you're under the effect of the shapechange spell, you use your proficiencies, including your proficiency bonus, except when the stat block of the new form has a modifier (proficiency bonus + other modifiers) that's higher for a proficiency you have. #DnD
(This also means the Proficiency Bonus doesn't have to be considered a class feature to be kept in Wild Shape.)