Hinduism isn't really a single religion though. Different regions have different practices, worship different gods and have very different practices. The closest analogy would be claiming all Chinese folk religions are a single religion.
The concept that it was was created largely in the late 19th century as part of rising Indian nationalism and anti-colonialism.
Yes that is correct, Hinduism is indeed very diverse and not understood as one religion, especially in the tribal areas. However, the majority of India drew much closer in practice and social structure when the Brahmins between the 2nd century BC and 2nd century AD began absorbing local deities and reinterpreting them according to the Vedic legends (the so-called Hindu Synthesis). The same Brahmins, by circulating between different parts of 10th century India, helped to create a sense of solidarity (Cynthia Talbot. "Inscribing the Other," 1995) In the 11th century when the Muslims came, these groups began to collectively refer to themselves as Hindus (the name that the Muslims gave them) and were collectively outraged by, for example, Mahmud of Ghazni's attack on Somnath. There were countless cases of Hindu-Muslim rioting coordinated across multiple regions of India before the British colonized the place, for instance, the Hindu rebellions against Mughal Padishah Aurangzeb.
Basic forms of Hinduism, where local deities are interpreted according to the Vedic stories and the priests adopt the name, language, and the rituals of Brahmins, is precisely what was found throughout SEA before Islam/the Spanish conquest of Philippines/the Vietnamese conquest of Champa. For one example, see this thread in PDF
on the Thais, a practice which originates in the Angkor empire.
Modi and his Brahmin oligarchy has very old roots and is the only thing that pre-British India has ever had that resembles a pan-Indian rallying point, even if it only covered half the population that actually believed in it. For decades Congress relied on Muslims and Scheduled Castes to shore up their voting bloc while co-opting local Brahmins, which has stopped working. Indian politics is likely to stay like this for some time, which will probably polarize India and encourage more separatism like the Khalistanis. The big prize is Dravida Nadu; BJP has little support in the south.