Using the term 'feudalistic' and 'slavery' as a comparative isn't a very appropriate way to view either system. Feudalism implies an aristocracy who owns land by inherited right, but also have a duty to the bonded serfs who work the land and recognition of a greater authority. Slavery implies the use of people as liquid capital, buying and selling their labor as capital and complete subservience to the owner. Serfs had the right to justice, allowed limited mobility, and could accumulate personal wealth (if circumstances allowed).
In terms of Tibet, the argument that it was a religious society based on a feudal system is accurate -- but it was accurate for a large number of nations in that region at that time, and was not out of the ordinary. Russia was, to a certain extent, a feudal society until 1917 as an example. Tibet was in a process of modernization starting in the 1930s; partly due to the external pressures of Nepal, India, and China encroaching on their territory. Due to their geographic remoteness, sustained trade was difficult, and the federal government had problems both with ensuring taxation and maintaining relations with Britain as a primary trading partner in the 1920s and 30s.
In short, Tibet effectively had a series of regional overlords who were accountable for local administration and governance. But there was a recognized federal authority and at the time of occupation Tibet was undergoing a series of initiatives to modernize and open their economy. Tradition, religion, geographic remoteness, and a weak economy all contributed to a slow transition away from a feudal government.
Reference material: A History of Modern Tibet, 1913-1951: The Demise of the Lamaist State, By Melvyn C. Goldstein
There's a lot of good information in this thread and I wasn't going to comment at first, but there's some inaccuracies and misrepresentations (minor at best). Still, I figure I'll throw in my 2 cents to get a bit of a bigger picture here.
The beginning of the Dalai Lama (used as a title here) rule in Tibet began in the mid 1500's. At the time Tibet was under the rule of the Mongol Empire. A Dalai Lama at the time established a close relationship with the Khans by declaring he was the reincarnation of an earlier monk that had converted Kublai Khan, and the current Khan of this sub-group was the reincarnation of Kublai Khan. This started a trend where Buddhism was popular among mongol elite, and the Mongols favored certain Buddhist leaders. There was a prolonged civil war between various sects of Buddhists, and one group eventually succeded. - u/BigBennP
The Mongol Empire here is only VERY loosely connected with the Empire that Chinggis Khaan founded in 1206. The North Yuan were still doing their thing after being bested by the Ming Dynasty in 1368. They were faltering but received a solid revival under the Great Khaan's descendants Manduhai Khatun and Dayan Khan at the turn of the 16th Century. This becomes somewhat relevant in that the above King and Queen would be the great-great-great grandparents of the Fourth Dalai Lama.
The Dalai Lama lineage officially beings in 1391 with the birth of the famous lama Gedun Drup. He was not acknowledged as "Dalai Lama" until centuries later when the title would be applied posthumously to him. At the time, Gedun Drup was simply acknowledged as the reincarnation of the famous teacher Lama Drom. He was also the star pupil of the great reformer Lama Tsongkhapa who was recognized as an incarnation of Manjusri, the Bodhisattva of Wisdom (just as Gedun Drup and the Dalai Lamas are recognized as an incarnation of Chenrizi, the Bodhisattva of Compassion).
Going back to this "Mongol Empire," it was really a piece of territory in western Tibet and southern Uyghuristan controlled by Dayan Khaan's grandson Altan Khaan. The Mongols for the most part were now heavily invested in Tibetan Buddhism. Going back to the great Qhubilai Khaan, who ruled the Yuan Dynasty after his grandfather Chinggis Khaan, who patronized Lama Pakpa. Pakpa was the ruler of the Sakya sect of Vajrayana Buddhism so his employment by the then-recognized King of the World was a hug boon for the Sakya sect who built monasteries all across Asia (as far as Russia and Persia, few of which survived the Ilkhanate or Golden Horde conversions to Islam). Centuries later, Altan Khaan asked Sonam Gyatso to recognize him as the reincarnate of Chinggis Khaan. Altan Khaan wanted nothing more (like many in central Asia at the time) than to conquer the world like his distant ancestor. But he needed that added legitimacy. In 1577 Sonam Gyatso, the grand-reincarnate of the above Gedun Drup, recognized Altan Khaan who then somewhat surprisingly recognized Sonam Gyatso as "Dalai Lama." "Dalai" being the Mongol direct equivalent of "Gyatso," both of which mean "Ocean" implying the Lama's wisdom is as vast as an ocean (a word which carries a lot of weight in landlocked countries).
The Dalai Lamas, however, were students of Lama Tsongkhapa's school, the Reformed Kadam which eventually was called the Geluk school. Lama Tsongkhapa and the first Five Dalai Lamas each founded and built monasteries across Tibet. (Tsongkhapa built Ganden, (HHDL I) Gedun Drup built Tashilhunpo, (HHDL II) Gedun Gyatso built Chokhorgyel, (HHDL III) Sonam Gyatso built Kumbum, (HHDL IV) Yonten Gyatso built... something, I need to go back and look it up, (HHDL V) Lobsang Gyatso built the famous Potala Palace). Obviously, if you build it, it becomes your seat until further notice. The Dalai Lamas, head of the Gelukpa school, now didn't have to worry about a school like the Sakya having grand imperial connections, had royal connections of their own, and now had a growing infrastructure of monasteries and temples. Until the Fifth Dalai Lama assumed temporal and spiritual control over Tibet in 1642, there was a "prolonged civil war" in the form that the Kagyu school of Vajrayana Buddhism competed (sometimes violently) for influence over Tibet with their main rival being the Gelukpas.
Lobsang Gyatso the 5th Dalai Lama (1617-1682) is known for unifying Tibet. Gushi Khan aided in making the 5th Dalai Lama the spiritual and political leader over most of modern Tibet. Tibet continued to be governed by the Mongols or various related groups until 1720 when the Qing Dynasty established a protectorate over Tibet and installed the 7th Dalai Lama as their puppet. Those states continued to lead through local Tibetan nobles The Dalai Lama would remain the de-facto leader of Tibet until 1962. - u/BigBennP
Gushri Khaan was a pious Mongol king who ruled over the Qoshot tribe that would later settle around the Lake Kokonor (Lake Qinghai) region where their descendants still live. The Fifth Dalai Lama's regent Sonam Rapten asked Gushri Khaan to help end the competition with the Kagyupa. Gushri Khaan took Lobsang Gyatso as his tsawa lama (root guru, primary teacher) and swept out the Kagyu order and seating HHDL V on the golden throne in Lhasa in 1642.
During the enthronement ceremony, it's important to note that the Dalai Lama was seated in the center significantly higher than the Khaan or the Desi (Regent) who sat on either side of Lobsang and were seated at equal height. The seating arrangement is important because it recognized the Lama's higher position over the Khaan (though there is significant debate over whether the Desi really was in control of Tibet's external affairs and how much the Mongol warriors had to play in Tibet's mosaic of society).
The government officially ran out of Ganden Goenpa - the Ganden Monastery that Lama Tsongkhapa founded above. Until the PLA toppled the Tibetan government in 1950, the Tibetan government actually referred to itself as the Ganden Phodrang. In addition to unifying most of Tibet under the Ganden Phodrang (with various levels of control in outer Kham and Amdo, and the rebellious kingdoms of Ladakh and Bhutan which broke away during the Fifth Dalai Lama's rule and became havens for Kagyupa refugees) the Fifth Dalai Lama was incredible at infrastructure building. He built medical colleges and clinics all across Tibet, initiated the first census, maps, and survey of the country to encourage effective government, built the first Tibetan treasury with the first organized system of taxation since the fall of the Tibetan Empire centuries earlier, and finally promoted a tradition of religious tolerance (somewhat ironic considering his rise to power involved sectarian wars)
The Great Fifth, as he is known in Tibetan circles, cordoned off some land in Lhasa to build a mosque for Kazakh traders. After all the violence with the Kagyu was largely over (in Tibet) the Dalai Lama (or his Desi) sought to end the Kagyu fringe by promoting local Kagyu lamas over Ladakhi and Drukpa (Bhutanese) ones. The Dalai Lama, much to his contemporaries' chagrin, was also a recognized Nyingma Terton (treasure-revealer) and is recognized as one of the "Five Confirmers." He actually writes in his autobiography, "Gelukpa hate me because they say I am Nyingma, Nyingma hate me because they say I am Gelukpa."
The Great Fifth left a big pair of shoes to fill. His successor, the Sixth, was completely uninterested in political or religious happenings and chose the life of a sexual libertine by his enthronement at age 18 (where he refused to be enthroned). Unfortunately for the Lamas who tried to pressure Tsangyang Gyatso to ordain as a monk like his predecessors, they couldn't take back Tsangyang's recognition as the Dalai Lama and he had access to all of Lobsang Gyatso's vast territories, wealth, and power in the heart of the people. Since he never took the vows of a monk, only those who thought he should be a monk became angry that he was out having sex and drinking all night. There was even an attempt on his life on one of these nights. Contrast that with the fact that there was a shortage of yellow paint in Lhasa when every girl he slept with painted her house yellow as a sign that she was chosen as Kundun's consort.
Political intrigue in Lhasa was run by Gushri Khaan's successor in Lhasa, Lhazang Khaan. Historians are still confused by Lhazang, with many of them claiming he was pious and well-intentioned, that he never meant to hurt Tsangyang and what followed was accidental and out of his hands. His wife, who was spurned by the Sixth's Desi, executed the Regent which led to the Mongols arresting the Sixth (after threatening to destroy the monastery he was in at the time and kill everyone inside). Tsangyang sat in a jail cell for some time before he was led to China where he died en route.
In 1708, Kezang Gyatso was born in Lithang and eventually enthroned as Tsangyang Gyatso's rebirth. The Qoshot were busy doing whatever they pleased in Tibet at the time as Lhazang Khaan struggled to find a replacement. He took a monk (today known as the Chakpori Lama) and named him the "Right" Sixth Dalai Lama and that the previous search committee had made a mistake. When the Seventh Dalai Lama took control of the situation in 1720, he asked the Dzungar tribe of the Mongols, operating out of Uyghuristan and Tajikstan, to oust the Qoshot. The Dzungars did so, briefly restoring order in Tibet. The Seventh eventually reformed the Ganden Phodrang which became the official government of Tibet, unchanged until the PLA invasion of 1950.
Cont'd because wow
There's a problem with the word "Feudal" because it carries a lot of baggage with it. It causes people to imagine a lot of things that don't necessarily match reality. It's better just to describe the history of Tibet.
The beginning of the Dalai Lama (used as a title here) rule in Tibet began in the mid 1500's. At the time Tibet was under the rule of the Mongol Empire. A Dalai Lama at the time established a close relationship with the Khans by declaring he was the reincarnation of an earlier monk that had converted Kublai Khan, and the current Khan of this sub-group was the reincarnation of Kublai Khan. This started a trend where Buddhism was popular among mongol elite, and the Mongols favored certain Buddhist leaders. There was a prolonged civil war between various sects of Buddhists, and one group eventually succeded.
Lobsang Gyatso the 5th Dalai Lama (1617-1682) is known for unifying Tibet. Gushi Khan aided in making the 5th Dalai Lama the spiritual and political leader over most of modern Tibet. Tibet continued to be governed by the Mongols or various related groups until 1720 when the Qing Dynasty established a protectorate over Tibet and installed the 7th Dalai Lama as their puppet. Those states continued to lead through local Tibetan nobles The Dalai Lama would remain the de-facto leader of Tibet until 1962.
In 1912 when the Qing dynasty collapsed, Tibet declared independence. There were a series of negotiations between the British, Chinese and various other parties to try to resolve these issues, but they mostly just kicked the can down the road. Tibet remained independent until 1951 when the Chinese re-conquered it.
Donald Lopez notes that Tibet at the time DID have a very unequal society. However, the Chinese also played up the feudal and slavery angles to justify their own invasion of Tibet to claim they were acting in the interest of the Tibetan people. Slavery may have existed, but was almost gone by the 20th century.
At the time much land in Tibet was held by a class of nobles. Their estates were granted by the government and were hereditary, but could be removed at will. Tenants had property use rights (usufruct) which they kept by fullfilling labor obligations for the landowners. This is essentially a form of serfdom, but the 13th Dalai Lama had reformed the system in the late 1800's. Serfs were obligated to work for their lords, but any serf who absented himself for three years could be re-classified as someone other than a serf.
On top of this was the Buddhist Monastic system. Various Buddhist Monasteries owned large tracts of land and supported themselves through that land. People of any social class could join the monasteries, and the monasteries were at least in part, meritocratic. Sons of lower class individuals did at times, rise to the very top of the monastic system and become Dalai Lamas.
Source in part: Donald S Lopez Jr., Prisoners of Shangri-La: University of Chicago Press, (1998)
There's some great info here about what Tibetan society was like, but I'd like to address the "presided over" aspect of your question. The 14th Dalai Lama was only 15-years old when he was formally enthroned in 1950, the same year that the People's Liberation Army threatened Tibet, forcing the Tibetan government to sign the "Seventeen Point Agreement for the Peaceful Liberation of Tibet" with the PRC. Though the young Dalai Lama remained the titular leader of Tibet, he wasn't really "in charge." He fled into exile in 1959, aged 23, when his situation became untenable due to the the 1959 Tibetan uprising. He was just a kid with a fancy title who never really "presided over" Tibet.