So I'm defending with Ahri, and a Hidden Temporal Breach. They move in and Ahri's defend trigger happens. They pass and I reveal Temporal Breach on my Ahri. Does the defend trigger happen again?

Yes — Ahri's defend trigger fires again.

Temporal Breach banishes Ahri, and Banishment is a non-board zone, so she becomes a brand-new game object (Rule 124). That new Ahri has never gained the Defender designation in this combat, so when she gains it during the Cleanup after Breach resolves, her "When I defend" trigger goes on the Chain normally (383.4.f.2.a). This is the same line as the Arcane Shift + Ahri ruling (FAQ #5362).

The key difference from "moving to base and back": merely relocating without leaving the board keeps the same object, so the once-per-combat check would hold — banish-and-replay is what resets it. Also fine here: a Hidden Breach's target must be at the battlefield it was hidden at, and Ahri is right there defending.