Another educator answered that she feels concerned extrapolating too much out of Disney films, and she is right about that. Profiteering, crowd-pleasing children’s movies are often taken too seriously from a literary point of view, especially nowadays. That being said, Mulan is set in a vastly different culture, and fiction can be a good jumping-off point for learning about foreign worlds.
Religion in the far east, and China especially, is more of a melting pot than an orthodoxy. In the movie Mulan, Hinduism and Sikhism are not particularly relevant. They are both religions rooted in India, Sikhism being an Eastern monotheism and Hinduism not entering into the events of Mulan. In China, the major religions are Buddhism, Taoism, and Confucianism, with shades of Legalism and Chinese Folk Religion. Some people follow one of these schools of worship very closely, but most Chinese people mix and match, believing in some hybrid of multiple doctrines.
Mulan leans into Chinese Folk Religion with its reference to ancestral ghosts. A lot of themes, such as the lucky cricket and the existence of dragons, are torn straight from the Chinese Folk miasma. Mulan herself exhibits some Confucian and Buddhist traits, striving toward self-betterment and seeking to help her family and her nation as much as possible. The idea of placing the family above the Emperor (since the family is a microcosm of the Empire) is also firmly rooted in Confucian teachings. The stringent, bureaucratic world Mulan is pitting herself against is somewhat more Legalistic.
Dissecting religious influence in Mulan is difficult, because so many different religious ideals mix together. But that in and of itself is very true of modern China.