Now the word Esoteric gets thrown around a lot in the martial arts world. Many people don't even know what the word esoteric means. So lets start there, generally speaking the term esoteric is related to religion, specifically speaking the Greek Mystery Religions. Esoteric generally refers to Private or Secret Knowledge belonging only to a select few. Now in the "ninjutsu world" this gets broken down in to two things, 1) strategy and philosophy that address incites into human nature and/or 2) religious mysteries and philosophical incites into human nature.
One good example of such case is the often debated Sakki element of martial arts training. Now Sakki is literally translated as "Air of Murder" and refers to the uneasy feeling one gets when danger is close by. Now there are multiple explanations of this, ranging from "guardian angels" and mystical powers gained through meditation to the subconscious mind and psychic powers. Now guardian angles is a general term for guardian spirits & not specifically limited to Judaeo-christian or Islamic mythology. Sakki can best be summed up in modern terms as a kind of "Spidey Sense" that warns of possible threat.
It is actually interesting that the scientific community has attempted to investigate and examine this phenomenon. On multiple occasions, and science has explained this (at least in part), psychologists and other researchers have found that the subconscious mind seems signs of a threat and relays this information to the conscious mind as "feelings." So what is such a mystery concerning this matter? One study conducted back 2008-2009 produced this result:
The characteristics identified as beneficial to IED detection came as little surprise to JIEDDO’s command sergeant major, Command Sgt. Maj. Todd Burnett, USA (no relation to Dr. Burnett). Command Sgt. Maj. Burnett has extensive in-theater experience, including performing route clearance missions, and he says he expected the person who would be best at finding the devices would be a little older and have hunting experience because that involves the ability to remain still and take in a 360-degree view of the surroundings.
So is hunting an esoteric practice? In ninjutsu there is a term Haragei which literally means "stomach art" and it is the Japan practice of reading (between the lines) non-verbal ques. It is not specific to martial arts but Japanese culture. You see the Japanese are very polite people and rather risk offending someone with direct response that would be viewed as negative, they give indirect responses that basically conveys the negative response in a less perceived offensive light. So must have a Haragei "gut instinct" for what is meant regardless of what is said. Understanding a true meaning through implication. This also means the Japanese deal more with reading body language and what isn't said as much as what is said... This not secret art form but a common cultural aspect of the Japanese culture.
In martial arts there is also the term Mushin which means no mind, and this has a simple meaning of acting without thought. Mushin is gained by developing one's Kan or intuition. There is a story in Shotokan of how the founder of Shotokan Funakoshi Guchin asked his students to attack him at random to develop his skill in Karate as a form of training. Following modern studies into soldier's Combat Intuition this a simply building said intuition through experience. This is good training and not some "secret" known only to a chosen few. It develops Kan or Intuition.
Now most often the "esoteric" teachings come in the form of quasi-mystical semi-religious thought, what child psychologists call "magical thinking." Now magical thinking is basically counter to any religious belief system, and I don't want to be accused of picking on anyone's religion so I will leave my religious beliefs to be addressed by addressed below. I will state religions are like relationships and so can be toxic, misleading and abusive. I know a great many people who have done more harm to themselves using the religion of Wicca to try to force things to happen or the reading of horoscopes and tarot cards to convince themselves of lies. It is of course not the fault of the Wiccan religion that they used "magical thinking" to deceive themselves.
The five elemental philosophies of Taoist Alchemy and Shinto are a philosophical approach to nature science from a far older time. Thus they approached the elements of nature at basic levels, water, fire, earth, air & nothingness. These elements became expressions of energies, ideologies, personality types (archetypes) and motivations. They are elements of an ancient science and part of cultures and religions. Like the Ten Commandments, they do not have to be linked to religion to have value. You do not have to believe in the Judaeo-Christian God or Jesus to believe lying, stealing and murder are wrong. Nor do you need to worship the gods of Buddhism and Shinto use the ancient science within the religion. You don't have to believe in "Ki" or "Chi" (a manifestation of the void) to believe in Sakki or develop Kan. Often times these practices are used to entrap the weak minded and those who feel powerless by offering a "secret power" or Red Badge of Courage to feel special.
Budo doesn't and shouldn't be seen as a mystical philosophy. Quite the opposite, the Way of the Warrior is very straight forward, although the methods are generally not. The Samurai often wrote poetry and examined or expressed philosophies. Reading the Hagakure: Book of Hidden Leaves we see that certain Shinto religious philosophies are carried over and defined Yamamoto Tsunetomo, a Tokugawa Era Samurai. It has many comparable philosophies to the Go Rin Sho (Musashi Miyamoto's Book of Five Rings). While the standardized Bushido is a more modern construct based on historical philosophies the oldest of which was the Kojiki, dates back to 8th Century and is mostly a collection of old myths and stories, the story of Prince Yamato Takeru provides a basis for early warrior standards based off his live and exploits.The Gunki Monomatari or War Tales puts a particular focus on warrior ethics and calls lists them as Loyalty to one's Lord, Courage in the face a certain death, Alliance to Family and Personal Honor tied to Loyalty. Other, such historical accounts such the Tale of the Heike (Heike Monogatari) focuses on a fantastic and romanticized view of the Genpei War (a 5 year war between the Taira & Minamoto Clan), these same virtues of Duty, Loyalty, Selflessness, Courage & personal Honor are focused on.
The term Do is the Japanese way of pronouncing Tao. Do is also linked to the Buddhist Sanskrit word Marga meaning Path. This frames the understanding of the meaning of Budo, as opposed to Bujutsu. Bujutsu is quite literally Military Art or War Art and Budo meaning Military Way/Path or War Way/Path. While art is generally a reference to the methods of war, techniques and strategies used in battle, way or path often speaks of spiritual, religious or philosophical path. So if we leave out the spiritual and religious path we are left with a simple philosophical path. A way of thinking and acting in accordance with a non-religious philosophical values system. This makes the value of Bushido (The Warrior Way/Path) and Budo (The War Way/Path or Way/Path of War) basically the same in concept.
If we apply religion, it is the religious interpretation of the warrior's path. Looking at history and Shinto is especial true in the case where the Imperial bloodline is said to be derived from the Kami. Loyalty to one's lord is essentially loyalty to the Imperial family and to one's divine beings of religion. Other virtues of the religion would of course also be associated with the virtues of a warrior. Certainly Shinto and Buddhism aren't the only religion to express this, even Christianity has it interpretation of the warrior's path. So what is so secret about the Warrior's Path that it conveys esoteric meaning? Does Honor, Duty, Integrity, Courage & Loyalty has some secret that only a chosen few can understand or is it just the quasi-religious cult following a martial arts cult?
The way of the warrior is straight forward, to charge headlong into adversity, conflict and hardship (combat both physical and mental) and the Way of War is embrace the Warrior mindset and character in mundane life and technical elements of military or paramilitary conditioning (martial arts) to develop the warrior character and teach warrior philosophy. In this way, Budo can be applied to business strategy and other aspects of life like other society interactions, not just a strict adherence to warfare.
Sometimes (more times then we would like to admit in the martial arts culture), Budo is used to teach and encourage Asian religions and other practices. This is often an interesting question when embraced by Atheists, Judaeo-Christian believers, Muslims and others who do not wish to stray from their ancestral or chosen religion. As it is a bit like tricking someone to converting to another religion, and many practices seen in martial arts training, like bowing a picture of the schools founder, then clapping and chanting a mantra are characteristic of said religious practices of Shinto and worshiping the "divinity" of said founder. This the problem with esoterics in martial arts...