Is Bastion "inherently harder" the second time?

Well, the kid is more powerful because of all the upgrades, and has access to more shrines to empower his enemies ...

Regardless of that - are the enemies in the second round harder, assuming all else is equal?

No, New Game+ is not inherently harder than the first time around. It's been a while since I played it, but as I recall the enemies aren't stronger or anything. You can just keep leveling up your character (and get the other ending without reloading a save).

New Game+ isn't inherently harder than the first run through the game. However, completing the game the first time unlocks the remainder of the idols available in the Shrine, which allows you to invoke all ten gods as you play back through.

As a design decision, the idea here is to allow players to choose their own desired level of difficulty on a per-level basis (ignoring Pyth Orchard, where you gain access to the Shrine for the first time). The difficulty modifiers directly impacts the amount of XP and Shards of the Old World you receive in battle, and they increase your score and multiplier duration in Score Attack mode.