I'm a yoga instructor! Yoga is a group of mental, physical and spiritual practices which started in ancient India!
How can we classify yoga? a spiritual exercises?
It definitely started as a spiritual exercise, but as of the 1980s, it became a popular physical exercise practice in the Western world.
Does each of the oriental religion like Hinduism, Buddhism and others  have their own form of Yoga?
Yes, they do! Hinduism, Buddhism and Jainism all have a broad variety of yoga schools, goals and practices!