[RPG] Can a mundane shield be used with Mage Armor

Armor bonuses are more common than shield bonuses.

Mage armor provides an armor bonus to AC, which doesn't stack with other armor bonuses. Since nearly everybody has an armor bonus (even proficiency with light armor gives access to mundane options that match the +4 bonus), this has a limiting effect on its usefulness to non-caster classes. This is why the spell is allowed to be cast on anyone and to last so long: it's not as useful so opening up the target range doesn't have as big an impact on balance.

Shield, on the other hand, provides the much less commonly-used shield bonus to AC (many fewer classes have this proficiency, even fewer of them usually take advantage of it, and getting a mundane shield with a +4 bonus is a build-defining choice). Very few classes run around with a +4 shield bonus, so even those who do use shields would find it very useful... and thus its availability is limited to the caster, to keep that usefulness in check.

It's the same reasoning that makes rings of protection (deflection bonuses) cost twice as much as bracers of armor (armor bonuses) with the same AC value: the less common a bonus is, the more valuable it is because it's less likely to conflict with another existing bonus (which would make it useless).

Two asides

1. It's worth noting that mage armor and shield are both force effects, which have various usefulnesses, but that only shield stops magic missile. It appears Wizards overestimated the value of direct-damage spells, and underestimated how easy ranged-touch-attack spells would be to land, so they thought magic missile would be a cornerstone of wizards everywhere. If that had been true, shield would be much more useful--making its duration/target restrictions more sensible.

2. Exactly how bonuses stack (especially as regards armor bonuses and enhancement bonuses to AC) seems to be rather murky and contradictory in the rules, so if further clarification is needed on that particular aspect of this issue, it should be made a separate question.

You can have two castings of the same spell active on you but you can only benefit from one at a time.

Per the PHB (pg 205):

The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don’t combine, however. Instead, the most potent effect—such as the highest bonus—from those castings applies while their durations overlap.

Emphasis mine.

When you have already lost the temporary HP from the first casting yet you are still under the effect of the spell, the second cast takes effect and grants you the new temporary HP. In this case, you benefit from the new casting of Armor of Agathys because it has a more potent effect (5 temporary HP) than the previous casting that is still active on you (no temporary HP).

Note that you are still under the effect of the previous spell until its duration expires (this might matter for dispelling or other things), but it has no effect because you've already expended its usefulness.