When you play Temporal Breach on a battlefield you control, does it start a showdown?

No — if the unit's owner already controls the battlefield, no showdown starts.

Temporal Breach banishes a unit and its owner plays it back to the same location. Contested is only applied when a unit becomes present at a battlefield its controller doesn't already control (Rule 190.3.a.1). Replaying your own unit to a battlefield you control doesn't satisfy that, so nothing is staged (matches FAQ #8066 — playing a unit to a battlefield you control never triggers a showdown).

One caveat: if you target an opponent's unit sitting on a battlefield you control, the opponent plays their unit back to a battlefield they don't control — that applies Contested and a showdown (combat or non-combat, depending on who else has units there) will be staged in the next cleanup.