My Shadow Clone attacks to a showdown, then I swap Zed Without a Sound to that battlefield instead and the showdown resolves. Zed trades 5 might into a couple of his units. My opponent still is holding that battlefield. If I play a card to move the Shadow Clone back into the battlefield to start another showdown, does the Assault trigger happen again? Would it attack for Assault 8 or Assault 4?

Yes — the trigger fires again, and with a second banish it's Assault 8 (Assault 4 if you can't pay the banish again).

Shadow Clone's ability has no once-per-turn limit — it's a "when I attack" trigger that goes off each time the token gains the Attacker designation in a Combat (Rule 807.1.d). Zed's swap moved the clone off the battlefield and the combat ended, which removes all Attacker/Defender designations (Rule 466.7.a). When you move the clone back onto a battlefield your opponent controls (and holds with units), it applies Contested, a new Combat Showdown opens, and your clone is the Attacker again — so the trigger re-fires and you can banish a second unit from trash.

The first Assault 4 was granted "this turn," so it's still active, and Assault values from multiple sources sum (Rule 807.2): Assault 4 + Assault 4 = Assault 8 for that second attack (FAQ #11782).

One caveat: if the re-entry only produces a Non-Combat Showdown (opponent has no units there), there's no Attacker designation at all — the trigger won't fire and the Assault 4 won't even apply, leaving the clone at 0 Might.