While Armadillo Cloak’s last ability is similar to lifelink, it isn’t lifelink—it’s a normal triggered ability. If a creature you control with lifelink wears an Armadillo Cloak and deals damage, first you’ll gain that much life for lifelink, and then Armadillo Cloak’s triggered ability is put onto the stack. This also means that multiple instances of Armadillo Cloak’s triggered ability do trigger separately.

If Armadillo Cloak enchants a creature you don’t control, you’ll gain life when it deals damage, as long as that damage hasn’t already caused you to lose the game.