This is a massive topic, and while this reply mostly refers to educated, scholarly medicine I can only begin to scratch the surface of Chinese medical treatment in this era (of which there were as many treatises as in Europe and just as much diversity of thought).

The model of the human body used in China during the Song and Ming Dynasties didn't quite have a name like "Galen-ism" or "Avicenna-ism"*, but there was a distinct model. It was folded into a general cosmological model of the universe, and was not considered fully separate from knowledge of the cosmos or even from sociology (though it was perhaps slightly less prestigious than more academic disciplines that didn't deal with physical bodies).

The Chinese model was based upon a schema that in English is translated as Song-Ming Learning. While the term "Chinese Traditional Medicine" is used in retrospect, that also is used to refer to a schema today that is descended from that. Authentic 13th century Chinese medicine is not identical to current alternate medicine practiced in the Sinosphere or marketed as Chinese, although there are of course commonalities. I recommend books with titles more like "Medicine and Society in Late Imperial China" than anything that uses the term "Traditional Chinese Medicine" unless it has a date or dynasty attached.

The most important aspect of this to grasp is what English calls "the Five Elements". A better translation of the Chinese Wuxing (五行) ('woo-shying' in English phonetic transcription) would be Five Phases. 'Elements' implies to the modern mind a concrete building material of the universe whereas the Chinese saw it instead as five linked sets of fundamental traits/behaviours/motions/phases of matter which anything could assume).

Each organ was associated with an element, which was associated with times of the day, physical matter, foodstuffs, tastes, senses, cardinal directions, et cetera. Certain medical procedures had elemental associations, such as burning incense being used in Fire-aspected procedures. Medical treatment was based on the idea of cycles of the elements: If someone has too much Fire, you should treat them with either more Water or Metal (the two elements that suppress fire). All aspects of the cure (time of day administered, materials used, et cetera) would be at least formally related to the Five Phases- you could not, or at least should not, prescribe in a vacuum.

The five phases were in cosmological and medical thought by this time integrated with the previously distinct cosmological theory of Yin-Yang which also had medical implications. Yin (inward, cool, feminine, receptive, earthly energy) and Yang (outward, hot, masculine, powerful, heavenly energy) are in Neo-Confucianist and Daoist thought of the time the two fundamental things which came out of existence in the primordial chaos. Each element could be both Yin or Yang (perhaps you are familiar with the Chinese birth signs, with things such as "Year of the Yin Fire Horse"), and respecting over-Yin or over-Yang conditions was just as important as the five elements. As Yin and Yang have distinct, gendered implications, they are important for the treatment of sexual health especially.

The universe, as I understand it, is instantiated in... "pattern" or perhaps "telos" (li) as "energy" (qi) (these translations are intended to give the proper feeling rather than be literal), two basic things similar to how we might talk about 'matter' and 'energy' or even 'form' and 'function' as fundaments of reality. Circulation of Qi, where Qi here signifies "[Vital] Energy", within the body was of paramount importance, and medical treatment often focused on 'removing blockages' or 'shifting concentrations of qi'. Acupuncture, with which you are undoubtedly familiar, is a treatment that is intended to prod critical areas in a body and remove blockages in qi circulation.

During the Song and Ming period the school of Neo-Confucianism was coming into existence, which attempted to apply Confucianism to cosmological and logical questions in addition to sociological and ethical ones. A common belief at the time was that of the harmony between the three powers- Heaven, Earth, and Man. It was thought that humanity was co-equal in a way to the heavens and the earth as a cosmic map, meeting the two in the middle- as Heaven is relatively Yang and Earth is relatively Yin, Man is a balance between the two and brings harmony between them. Man has a heavenly spirit but an earthly body, et cetera.

There were many folk beliefs or religious beliefs that intermeshed with this model. One of the more famous, but one I must stress was not an overwhelming obsession of medicine but a side point of one religion: Some sects of Daoism believed that male Yang life force was partly embodied in the semen, and that a fixed quantity of male life force was available to any person. They believed that the fires of a man's Yang could be stoked by sexual activity, but ejaculation would drain his life force, and so attempted to do the former without the latter. This also was an explanation against homosexuality- both partners lose Yang energy from ejaculation. Female homosexuality would not cause this problem- the receptive Yin wouldn't be receiving anything, but neither would it suffer. I don't believe this fostered any tolerance of it as much as it was simply considered irrelevant to medical treatment.

Folk cures or known non-systematic cures ended up incorporated into the elemental model. If a plant was reputed to have a use in treating Pneumonia, it was explained in the elemental model with "Pneumonia is a watery coughing illness. This herb treats it, so it must have a strong Earth nature to suppress the Water. Consequently, it is contraindicated in diseases of a Metal sort." This is not necessarily to say these cures were wrong or spurious, but the reasoning behind explaining why they worked was alien to western medicine.

The mind and body were not regarded as two separate entities in a cartesian sense- it was recognized that one influences the other and vice versa. A person's medical treatment would consider their emotional states for elemental criteria- an excitable, loyal person would not be provided fire-aspected treatments if possible or it could theoretically cause further illnesses by promoting too much fire. It's important to mention that this itself could then produce even worse things by the fire causing a cascade- too much fire can make too much earth and too much earth can suppress too much water and so forth. Chinese medicine saw the body as in a careful elemental and yin-yang balance, and treatments were meant to restore it without any overcompensation.

Some folk remedies, even those used by highly educated alchemists and medicine-workers, were actually highly harmful. For instance, Mercury was thought to be a wondrous magical substance that could be utilised in a variety of ways, potentially even to obtain immortality in certain sects of Daoism. In Daoism (but not as much in Confucianist thought) mortality was thought of as an exorcisable flaw in humanity, if only the proper medical procedures and mystical disciplines could be performed. Mercury based medicine and Arsonous Oxide, amongst other outright toxic compounds were thought to be medically helpful. There was however an undercurrent of thought in Daoism at the time that mercury based immortality potions and other methods were harmful or dubious, as immortality-seeking turned from external concoctions to the idea of "Internal Alchemy" to try to produce longer life or spiritual survival after death or even permanent physical life. In Daoism especially what we would nowadays consider outright magical treatment blended in to what is recognized as medicine, with things such as incantations or talismans being used alongside herbs and physical procedures. (There was no such distinction between "Magic" and "Not Magic" back then in the same sense).

Like in Europe at the time, Empirical Testing was thought to not be as important as philosophical aptness and the ability to derive things from first principles. While the functions of the organs were known (generally- Chinese traditional medicine believed the kidneys manufactured gametes and that the spleen somehow provided energy necessary for function to the muscles, amongst other beliefs), treatment did not consist of prescribing a medicine and keeping careful statistically valid notes on what happened. Anecdote and theory ruled the day, as they did everywhere (as late as the 1860s I know that even European medicine could still sometimes scoff at the idea of numbers and testing replacing instinct and theorising**). Evidence based learning became more popular in the late Ming and early Qing dynasties after criticism of the Song-Ming era's overreliance on theorisation and categorisation, but even then the hard-nosed empiricism that characterises modern medical science was absent.

I know too little of the Mongol era, although I have not heard of any changes in medical treatment or the cosmological and religious thought upon which it was based. All that I can note is that the fact that Song-Ming Learning brackets both ends of the Yuan dynasty is rather suggestive that there was continuity in medical theory during it.

*There were several "Name-ism" schools as subcategories of thought based around beliefs of the proper way to treat illness and produce good health and welfare, but I do not know enough about them to summarise their differences. They all to my knowledge followed the general outline of this, although Neo-Confucianists and Daoists and Buddhists would have stressed different aspects and differed on some aspects (one fact I know for certain is that many forms of Daoism differed vociferously with one another and with Buddhism on sexual health.)

**See the tragic story of Ignatz Semmelweiss