What happens to breastplate Armor when it is chosen as Armorer Artificer’s Arcane Armor

Right now, they're the same. But at some point, you're probably going to want to raise your Dexterity up to 20, and at that point, the studded leather will be worth 17 AC (12 + 5), whereas the breastplate will still be worth 16 (14 + 2).

More generally, as you say, there's no reason for a piece of medium armor and Light armor to be statistically identical. But in general, they're not. They're identical for you, currently. For a character with 14 Dex, the breastplate is superior. For a character with 20 Dex, the studded leather is superior.

It's up to the DM to use common sense.

"Use common sense" is exactly the guidance given in the Dungeon Master's Guide (p. 141), under "Multiple Items of the Same Kind":

Use common sense to determine whether more than one of a given kind of magic item can be worn. A character can’t normally wear more than one pair of footwear, one pair of gloves or gauntlets, one pair of bracers, one suit of armor, one item of headwear, and one cloak. You can make exceptions; a character might be able to wear a circlet under a helmet, for example, or be able to layer two cloaks.

The Armorer Artificer's Arcane Armor feature says, in part (TCoE, p. 15):

The armor attaches to you and can’t be removed against your will. It also expands to cover your entire body, although you can retract or deploy the helmet as a bonus action. The armor replaces any missing limbs, functioning identically to a limb it replaces.

To this DM, it looks like you would be functionally wearing boots and gloves, since the armor covers the whole body. As for a helmet, it seems there should be no issue wearing another helmet while your Arcane Armor helmet is retracted, just be careful not to engage your helmet while wearing another one.