Does the magical bonus of armor and a shield stack?
For example, would a PC with a +2 armor and a +2 shield be able to stack both bonuses?
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In D&D5 everything with a plus/minus stacks with every other thing with a plus/minus.
Things that set a value do not stack with other things that set a value but do stack with things that give a plus. For example, the Mage Armor spell sets your AC to 13 + DEX bonus; it does not stack with a draconic sorcerer's natural armor which also sets your AC to 13 + DEX bonus, however, either will stack with the Shield spell which adds +5 to your AC.
Normal armour stacks with a normal shield, and there's nothing in the rules to suggest that this is different for magic items. It even allows you to wield/wear multiple of the same item, for example three rings of protection, granting you a total bonus of +3 to AC.
The only mention of multiple items with the same effect is the "multiple items of the same kind" section on page 141 of the DMG. This section states that you should use logic to decide whether a character can wear more than one of a given item. There's nothing in the book that suggests that multiple rings of protection cannot be worn at the same time (or even multiple cloaks of protection, for that matter), so if that's the case, there's no reason why you shouldn't be able to get the magic bonus from a magical armour and magical shield.