How does the Japanese religion of Shinto contribute to the cultural significance of this mountain?

Many Shinto temples are built directly on the mountain itself.

The mountain is considered sacred by the Ainu people, the indigenous inhabitants of ancient Japan. In fact, Mt. Fuji's name is derived from Fuchi, the Ainu god of fire and the hearth.

Some Buddhist sects considered the mountain a holy place, and from the 12th century CE, it became a destination for those practising asceticism (shugendo) and seeking a 'rebirth' from their time on the mountain, in a mix of Buddhist, Taoist, and animist beliefs. 

In fact, eight major shrines were built around the foot of the mountain and hundreds of smaller ones have since been added. The most important Shinto shrine is the Fujisan Hongu Sengen Taisha which was first constructed in 806 CE, although by tradition, it was originally founded during the reign of Emperor Suinin (29 BCE - 70 CE) at another location at the foot of the mountain.