No. Armor Class is not an ability check.
The PHB does not describe armor class(AC) as an ability check. Armor class is its own in-game thing.
Armor protects its wearer from attacks. The armor (and shield) you
wear determines your base Armor Class. (PHB, p. 145)
While the Dexterity ability adds or subtracts to AC in some cases via the Dexterity modifier, it is not universal:
If you wear heavy armor, your Dexterity isn't added to your AC.
If you wear medium armor, there is a limit (2, or 3 with the Medium
Armor Mastery feat) to how much dexterity modifies your AC.
If you wear light armor or no armor, your full Dexterity modifier is applied to determine your armor class. (Armor Table: PHB, p. 145).
The game term used is Dexterity modifier, not a Dexterity based ability check. Further that point, your AC is generally a static value (with specific exceptions depending on feats, cover, magic items, using your shield or not, etc. as you noted). You don't roll a d20 to determine your AC, whereas for an ability check you roll a d20 against a DC to determine success or failure.
Ability Checks (PHB p. 174)
An ability check tests a character's or monster's innate talent and
training in an effort to overcome a challenge. The DM calls for an
ability check when a character or monster attempts an action (other
than an attack) that has a chance of failure. When the outcome is
uncertain, the dice determine the results. (PHB p. 174)
@T.J.L. made the following point in a comment that may aid in understanding:
Ability checks, Attack rolls, and Saving throws all involve rolling a d20+modifiers. AC, while not a static value, is (1) not a bonus and (2) is never part of a roll: it is a target that a roll tries to meet.
If you are trying to view AC vis a vis ability checks, AC more closely resembles a DC(Difficulty Class) that an ability check tries to beat.
