The primary defining characteristic of a Role Playing Game or rpg is some sort of noticeable progress in character strength and abilities sometimes, most often actually, it's called a leveling system but not all rpg's have one. Some rpgs prefer to give you all your abilities at once then find new and creative ways for you to use them. While you might jump to calling zelda a fantasy action franchise I would suggest based on how your character levels gaining health abilities and certain items in the majority of the franchise it fits the bill for an rpg. The more important thing to note is that while the action and fantasy elements are well balanced there's a clear and definitive lean towards the story rather than the action again pushing me in favor of labeling it an rpg.
The greater majority of rpg's don't focus on heavy action, player reaction times, or physical skill rather most combat is tactical. In turn based or table top games you're working with a set of stats are you're calculating probabilities really where in comparison to real time action similar to zelda it's usually about finding the weak spots for critical damage. Think on ocarina of time how you really could only defeat the bosses in one specific way for the most part. Generally speaking an RPG is mostly about a small group of individuals or a single adventurer on a quest or adventure and the primary functions are related to the player or the story such as inventory management, dialog options in the storyline, and similar details.
These attributes remain across the board whether it's an action RPG, a western style, a JRPG or any other form. Though it's true many of these core aspects have been adopted into other games it's important to know where they come from and what their original purpose was which is immersion. The RPG is known for detailed stories and characters regardless of the size of the world the fiction was always vast and rich. It's not often you're so engrossed with your characters and their allies that when one of them dies it can emotionally touch touch you, and yet for the games that do more often than not they are RPGs. For every action game or shooter there are twice as many rpgs that have moved you, the entire genre is designed to attach you with your characters so that their quests becomes yours. Designed to make you emotionally invested so that every dramatic moment is like it's actually happening to you, a fact made easier by the overwhelming amount of fiction created to immerse you in the world.
The principal behind this is you're having a new and novel experience which your mind can absorb rapidly and as it's getting lost in the detail it uses constants like your characters as a point of reference and a sudden loss of that can be dramatic while also causing rapid firing of neurons as they panic to accommodate the change often hitting various centers of the brain near emotional controls. Truly RPGs are the most finely honed genre when it comes to player attachments.