I don't know much about Hinduism and the details of Indian history, but I feel that in India it is very common to nationalize foreign religions, starting with the indianization of Sumerian deities, then with the indianization of Taoism that resulted in Buddhism, and then with the indianization of Islam which resulted in Sikhism. I thought this was hidden due to religious and nationalist issues but then I read that the Advaita Vedanta states that the same truth is described in different ways in all religions, that they all lead to the same end but were mixed with non-truth, and therefore they must extract this truth present in all religions and mix with their own beliefs.
So my question is, this idea that all religions are true is shared in others Hindu schools? Do Indians openly adopt and syncretize foreign religions?
I would disagree with some points you've raised in your question.
For one, Buddhism is not an indianization of Taoism. The early strains of proto-Buddhism are reflected in the Vedic literature, long before an established cultural link with China. Buddhism arose in the 6th century CE BCE, at a time of great cultural and political churn in the Gangetic plains. Within the span of a single generation, three great religious traditions (Buddhism, Jainism, and Ajivikism) emerged that differed from orthodox Hinduism on the central tenets of the nature of the soul, divinity, individual agency, and karma. These religious traditions were referred to collectively as the shramana ("one who labours", ie, an ascetic), due to to emphasis on asceticism in these traditions. Within mainstream Hinduism, these same questions are pondered upon in the Upanishads, composed around the same time. For ancient Indians, these traditions were not necessarily seen as mutually exclusive.
Nor would Sikhism be considered "Indianized Islam". A syncretic tradition, sure. Islam and its indianized form had been present in northern India for half a millennia before the advent of Sikhism.
I am unaware of any Indianized Sumerian deities.
As to the question itself, there isn't reason to believe that Indians adopted and indiginized foreign religions at a higher rate than any other people. Indeed, there were many religious traditions that arrived in India and left nary a sign. For example, the Greeks. We have evidence of the Indo-Greeks adopting Indian religions (mostly Buddhism), and none of Indians adopting the worship of the Greek pantheon. The same goes for the Scythians, the Huns, the Hepthalites, and the Kushans.
While advaita Vedanta advocates a universality towards divinity, it is also highly critical of other traditions. The other schools of vedanta are even more critical, and are very clear about the paramount nature of their particular deity.