Western philosophy has a long and broad tradition. It doesn't surprise me that in a first course in the West you would only be introduced to the major Western philosophers - but perhaps it is worth mentioning that there are other philosophical traditions out there which are worth looking at for their own intrinsic interest as well for the impact they have made on the West.
As an aside and a concrete analogy: one wouldn't for example be able to fully understand Western art without also looking at the influences coming from outside the Western sphere, like Incan and African art which are non-representational. One could make a strong case that the advent of non-representational art in the West was determined by the impact of non-Western art itself on the West.
It only takes a little reflection to understand that non-western culture will have their own modes of philosophy - simply because the questions of life and the world are common to all of humanity and not just the West. In other traditions they may be presented as religion, theology, mythology as well as the kind of discursive tradition that is presented as philosophy in the West.
In the West, because of the recent (since the enlightenment) secularization and desacralization project in elite intellectual circles, it's not usually realized how connected the Western tradition is to the form of their religion - Christianity. Two millennia of Christianity will have left its mark. There are obvious figures like Pascal and Aquinas, but Kant and Hegel in part philosophize because of religion.
Eastern traditions have left their mark on Western philosophy although one has to go looking for it. Schopenhauer, for example, said that the Upanishads were the consolation of his life. Simone Weil, Christian mystic and moral philosopher knew Sanskrit. Oppenheimer, administrator of the Manhattan Project quoted from the Bhagavad Gita, 'I am become death, destroyer of worlds', at the first nuclear test explosion.
They have also left their mark on popular culture from Zen and the Art of Motorcycle Maintenance, Yoda in Star Wars, the Beetles taking up with a guru, the films of Bruce Lee and Kung Fu, the 70s TV serial, and on high art through the paintings of Rothko and some of the short stories of Salinger.
There is also possibly a more hidden tradition which has been documented and argued for in McEvilley's The Shape of Ancient Thought, in brief, that there were philosophical exchanges from antiquity between India and the West. This is in all probability correct. After all, there is the Greco-Indo city of Ghandara in Afghanistan and one only has to look at the statues there to discern the merger of both influences.
As to why these influences aren't more discussed and why the philosophies elsewhere aren't given their due comes down to the intellectual parochialism and the soft power of cultural (for want of a better word) imperialism. The West is the centre and thus spreads its influence much more than it is influenced by.
I suspect as China and India become more culturally self-confident and learn to reinvigorate their own traditions rather than merely copying the West that will change.
Eastern religion is not just a philosophy. It is also a religion, and religions are a way of life. That is, they have practices, rituals and temples. In some places bhakti (devotion) is much more common than self-conscious philosophizing.
One thing is true - the discursive tradition of the East is played down in the West. Amartya Sen, in his book, The Argumentative Indian, tried to open that discussion up a little. One example will have to suffice:
In the West, the axiomatic system is said to have originated with Euclid, hence axiomatic systems are associated in the Western mind with mathematics - and thus the apocryphal story that Plato hung over his Academy 'Let no-one ignorant of Geometry enter here'. This axiomatic system was a secret treasure of the West which it has recently bequeathed to the rest of the world.
But in India the axiomatic system is associated with Panini, the Sanskrit grammarian, and philosophy there in the beginning had a deep relationship with language rather than mathematics. When one considers how important grammars are now in computer science, one can see the difference is not all that great. It's also worth noting that Wittgenstein inaugurated the linguistic turn in Anglo-American philosophy and that Saussurean linguistics was responsible for semiotics as well as Structuralism in Continental Philosophy.
It's likely that in both traditions it was the example of law that lead to the idea of axiomatic systems. But this is simply an educated guess.