Neither. Monotheists have one god, polytheists have several. Buddhists have no god(s)
Legalism, an ancient Chinese philosophy, is primarily focused on strict laws and governance rather than religious beliefs. It does not emphasize the worship of deities or a specific god, which makes it neither strictly monotheistic nor polytheistic. Instead, it can be seen as a secular philosophy that prioritizes order and control in society over spiritual matters.
The most common synonym used for polytheistic is "pagan", but this is generally seen a pejorative.
Hinduism is polytheistic, though some may argue that there are two levels of deities within the religion, which separate a unified, one "God" and some lesser, but equally divine "gods". Zoroastrianism is ditheistic (two gods --> Ahriman and Ahura Mazda). A lot of people would also vouch that Buddhism is polytheistic, in the sense that they believe in devas, souls that reach an almost god-like enlightenment, though Buddhism inherently is atheistic. Shintoism is also polytheistic, given that they believe in the existence of "kami" which are higher beings of spirit-origin. Not to get into any philosophical discussions, but if you were to break it down, even religions like Christianity, which believe in the unified God in simultaneous multitude (like the Trinity or the Virgin Mary's "redemptrix" status), the feminine "huri" element of divinity seen in the Bahá'í Faith and other Abrahamic religions (Islam, Judaism), or any other religion where God has multiple instances of being, you could raise the argument of polytheism or polytheistic references made to a given monotheistic religion.
Hinduism is considered polytheistic because it acknowledges multiple deities, each representing different aspects of the divine. However, it is also seen as monotheistic because all these deities are ultimately believed to be manifestations of the one supreme reality, Brahman. This concept of a single, all-encompassing divine essence unifies the various deities and traditions within Hinduism.
Buddhism is not a theistic philosophy. The existence of god or gods is seen as non-important as each of us must find our own path to enlightenment. If there are beings who would be classified as god(s) they do not interact with humans.
Christians are monotheistic. Some opponents claim that they are polytheistic because of how Christians believe in the Father, Son, and Holy Ghost. This however is a misconception as the Bible frequently states that God is one and that ye shall have no other gods before Him. The Figure of Jesus is God the Son, as He Himself stated many times that He was one with the Father. This was so eloquently testified upon Him saying "If you have seen me, then you have seen the Father". Also consider how John stated in 1 John 5:7- For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one.
There is no god in Buddhism & Buddhist leaders vary depending upon the different schools of Buddhism. Often people believe that H.H. the 13th Dali Lama as the leader of Buddhism, but that is only due to the fact that he is seen more often than other Buddhist leaders.
it was seen as the only alternative to Islam
There are many different religions. They can, however, be classified to be part of four major groups: -Abrahamic (or Semitic) religions, including Christianity, Islam, Judaism and their off-shoots (including possibly Bahaism). -Indic religions, including Hinduism, Buddhism, Sikhism, Jainism and other groups (including many new religious movements headed by a Hindu-ic guru, such as Osho). -East Asian religions, including Taoism, Confucianism, Shinto, Tenrikyo, Chondogyo, Xian Tian Tao (includes I-Kuan Tao) and other groups. Abrahamic religions are monotheistic. They worship the god of the Jews, known as Yahweh. They believe in heaven and hell, angels, demons, etc. They generally attest to the veracity of the Hebrew Bible - the Jews add a variety of other books (including the Talmud), the Christians add the New Testament and the Muslims add the Quran (which is said to be an "uncorrupted" Bible). They all venerate the Jewish prophets. Indic religions can be monotheistic, polytheistic and even nontheistic! Buddhism generally falls into the latter category, but there are many theistic Buddhists. Many Hindus claim Hinduism as a monotheistic religion in that all of the gods are only different incarnations of Brahman, but most people in generally see it as a polytheistic religion. Jainism is the same as Hinduism, while Sikhism is monotheistic due to Islamic influence. The Indic religions all believe in reincarnation (or rebirth) and karma. All of these religions are called "Dharmas" - they are considered to be the "Way" to escape Samsara. Jains, Hindus and Buddhists believe in moksa or nirvana, but Sikhs believe that one ascends to the Abrahamic heaven when you become a Sikh (non-Sikhs have to die and wait to become Sikhs in a later life to get to heaven). In Buddhism, nirvana is more like "cessation", or death. The East Asian religions are as theistically diverse as the Indic religions. Within Taoism it's not very clear, although Taoists do venerate many "immortals" and other greater beings, parts of the Tao Te Ching can be interpreted as monotheistic. Confucianism is generally seen to be nontheistic, although Confucius did believed in a "Tian" deity (well, more of a supernatural force, it's sometimes translated as "Heaven"). Shinto is the shamanistic religion native to Japan (it has Confucian and Buddhist influence), it is definitely polytheistic. All of these religions believe in the "Tao", roughly translated as the "Way" or "Path". Confucianism and Taoism, as with Mahayana Buddhism, are heavily mixed in with the so-called "Chinese folk religion".
Although often seen as an Asian religion, Buddhism is practiced almost everywhere in the world
We speak of monotheism when a religion only has one single god. Hinduism recognizes several gods. Even if one of those gods is considered the principal god, we still call it polytheism, not monotheism.
At the time of Buddhism's inception, Hinduism had a caste system that held priests ("brahmin") in very high regard. Buddhism, as well as Jainism, challenged this system by declaring that peoples' worth are decided by their actions, as seen in the Dhammapada, Chapter XXVI (The Brahmin).