While the level 4 fire shield spell is out and equipped, does it count as a shield and give the shield AC bonus? It isn’t specified in the description of the spell, so I’m just curious.
Fire Shield does not give an AC bonus.
As you say, the description of the spell does not specify that it is a shield that gives an AC bonus. Because spells only do what they say they do, no more and no less, the spell does not give an AC bonus.
Additionally, the description of the spell states that it's more of a body-surrounding aura than a sturdy, holdable shield:
Thin and wispy flames wreathe your body for the duration...
As Icyfire said, spells do only what they say they do. But more to the point of order: Many spells have Misleading Titles, and for decades, several spells were printed that began with disclaimers "this spell doesn't actually___ but".
For example Summon Wind Dragons from Al Qadim begins: "Despite its name, this spell does not summon any sort of creature".
The "Shield" aspect of the spell is metaphorical, in that it protects the caster, providing resistance to your choice of fire or cold.