When a unit gains the Attacker designation mid-combat through an effect like Nidalee, Cat Form's Ambush, does Demacian Standard's attack trigger fire again for that newly attacking unit?

Yes — Demacian Standard's ability is an attack trigger, so it fires again for the new attacker.

Standard reads "When one or more friendly units attack here, give one of them +2 [M] this turn." That's keyed to the Attacker designation, not to combat starting (Rule 383.4.e), so Nidalee ambushing to that battlefield and gaining the Attacker designation during the following Cleanup is a fresh trigger event — you get another +2 [M] to hand out, on top of whatever the Standard already gave earlier in this combat.

What limits it is per-unit, not per-combat: Attack triggers are only checked the first time each unit gains the Attacker designation in a given combat (Rule 383.4.e.2.a). So bouncing Nidalee out and replaying her, or re-designating the same unit, won't re-fire it — but a genuinely new attacker will.

One caveat: this only works if it's a real Combat Showdown. If Nidalee were played to a battlefield with no enemy units, no Attacker designation is ever granted, so nothing triggers.