The popularity of Studio Ghibli ultimately boils down to the spirit, aesthetics, and sensibilities of Hayao Miyazaki expressed through their respectives story and visuals. Not every Ghibli work is created equal, the ones where he is are quite memorable to audiences.
Filmaker Asher Isbrucker offers an interesting take on Studio Ghibli's appeal in his video-essay “The Immersive Reality of Studio Ghibli”, that the studio’s biggest asset to its lasting appeal of their so-called “immersive realism”.
While you may think that elements such as parents being turned into pigs (from Spirited Away), or a story about a warrior princess raised by wolves (from Princess Mononoke) and or even one about WWI fighter-pilot that's a pig (from Porco Rosso) may not be instill what some would regard as "realism". The collaborative effort and vision of the director and their animator’s ability to breathe like these fantastical narratives to make them feel through their passion and detail give Ghibli movies their appeal. The world-building done allows audiences to suspend their belief and buy into the Ghibli-created world where even the most imaginative and fantastical elements become feel normal and common place.
Finding a balance between fantasy and reality is often times very tough. The writers along with the animators develop even the most basic and mundane elements from scratch, with an ever so particular attention to detail, no matter how far fetched or mundane it may be, it still feels tangible and real to the audience. Such realism is not made to a tee. It's not a complete facsimile of real-life but rather an analogue of it where rules can be bent and broken to create fantasy. If something feels too real or too emulated it can easily feel fake to an audience. This is where the animators really show off what they can do.
Roger Ebert once commented in one of his interviews with Miyazaki:
I told Miyazaki I love the "gratuitous motion" in his films; instead of every movement being dictated by the story, sometimes people will just sit for a moment, or they will sigh, or look in a running stream, or do something extra, not to advance the story but only to give the sense of time and place and who they are. "We have a word for that in Japanese," he said. "It's called ma. Emptiness. It's there intentionally."
In Starting Point, Miyazaki explains his underlying philosophy:
Anime may depict fictional worlds, but I nonetheless believe that at its core it must have a certain realism. Even if the world depicted is a lie, the trick is to make it seem as real as possible. Stated another way, the animator must fabricate a lie that seems so real, viewers will think the world depicted might possible exist...
Miyazaki also mentioned that "animators are themselves actors". These animators need to consider the various aspects character’s motivations, and considered their unspoken mannerisms and expressions as well. These minor details provide a connection with the audience and makes them relatable.
You can see this things like the animated expression of movement, emulating the semblance and familiarity of how how we perceive our physical world the video-essay notes things like Kiki’s bow blowing in the wind (from Kiki’s Delivery Service), the flickering lights of the Catbus (from My Neighbour Totoro) or the heavy-feeling movements of the hulking insectoid creatures of Nausicaa of the Valley of the Wind, or even how Chihiro from Spirited Away puts her shoes on, noting how she takes her time to tap them, make them comfortable, then set out like an actual girl in real-life would.
Not only that but the detail and effort made to scene establishes the setting very well and makes the setting feel all the more real. The video-essay notes in particular how the bathouse in Spirited Away depicts various job posts, areas for sleeping, an even the varieties of soap and countless other details that make each seems to have their own story on top of the main story unfolding with Chihiro. Another noteworthy scene mentioned is the contrast of labor roles by gender of Irontown in Princess Mononoke, such as what the women do while the men are out gathering resources.
Aside from the visual appeal of the art and animation, the story aspect of Ghibili works are split across a spectrum of fantastical, simple and relatable, and nostalgic elements with mature themes. At the heart of most Ghibli series is a coming of age story, but the trope is not played off a novelty as you'd typically see with anime. These stories do very well in empathizing with its audience universally understood ideal and emotions of it's world and characters through a combination vivid expressions and fantastical elements in a way that can only truly be expressed through an animated medium. The studio does it in such a way that subverts what audiences typically expect from an animated work through its emotional and impactful expressions. It at times it can reminds us of who we are as humans and to be modest, industrious and respectful despite the strife that may come our way. Such empathy and imagination is likely what persists the popularity of Ghibli works.