Is re-equipping a gear to the same unit allowed

No — not while it's still attached. Once it's detached, yes, you can re-equip it to the same unit.

While a gear is attached, its printed rules text (including its Equip ability) is Inactive, so you can't pay the Equip cost to re-attach it (Rule 718.2; FAQ 730). But Equip targets "a unit you control" with no restriction against a unit that already held the gear (Rule 818.1.c.2). So if the gear is detached — by an effect like Strike Down or Veiled Temple, or by the unit leaving the board so the gear is Recalled to your Base — its Equip ability becomes active again and you may re-equip it to that same unit (or any unit you control) during your Main Phase in an Open State.