HA does have a codeshare with AA for interisland HA flights so they are friendly (although they do compete on some mainland flights). Think your luck with Admiral's Club entrance is probably more of an informal agreement. My father used to get into airline clubs in airports that didn't have an affiliated club (think he had Delta, USAir and the old TWA club cards) by asking the gate agent nicely and they would call ahead to the unaffiliated club to allow entrance. Also in some airports, lounges are shared between airlines (HNL AA club is also Air New Zealand, etc.).
Normally get into clubs myself using a Priority Pass card (good at many, but not all, lounges) or with the occasional freebie lounge pass.