Does welding a shield count as Armor for the Fighter's Fighting Style: Defense: "While you are wearing armor, you gain a +1 bonus to AC."?

I’d rule no. You don’t wear a shield, you wield it. 

https://www.dndbeyond.com/sources/basic-rules/equipment#ArmorandShields

Armor Class (AC). Armor protects its wearer from attacks. The armor (and shield) you wear determines your base Armor Class.

Options are also limited for situations in which a fighter would not be wearing armour but still have a shield.
One option is a barbarian/fighter with high enough dex and con to warrant going unarmoured.
In this case options for fighting style could include great weapon fighting ...

If a fighter had no armour but for a shield and had a defense fighting style, I wouldn't dispute a +1.

While typically an armor is worn and a shield is welded, RAW a shield is a subset of armor and the Defense Fighting Style doesn't distinguish between the armor you wear and a shield you wield as some other game features do. 

The Sage Advice Compendium official ruling  has a Q&A regarding shield and armor;

Can you use a shield with mage armor? Mage armor works with a shield. Shields are grouped with armor in the equipment rules in the Player’s Handbook, but various game features distinguish between the armor you wear and a shield you wield. Take a look at the monk’s Unarmored Defense feature and compare it to the barbarian’s version. In the monk’s version, you must both forgo wearing armor and forgo wielding a shield if you want to benefit from the feature, whereas a barbarian must only forgo wearing armor.

While not official ruling, a Dev's tweet more specifically address the question and clarify it's not intended to combine Defense Fighting Style with a shield alone;

@Gargamoyle does a shield count as wearing armor? barbarian unarmored defense and defensive fighting style?

@JeremyECrawford The Defense fighting style is intended to work with a suit of armor you wear, not with a shield alone.