I am more and more coming to align myself with the German scholar who has been claiming that we should accurately consider a 'family' of religions, as we think of Judaism, Christianity and Islam, when we think of Indian belief systems. According to him, the Vaishnavite school shares a theogony with others, but its prescribed path to salvation is so distinct that it should be seen as a distinct religion, its votaries should be seen as a distinct priesthood, its followers as a distinct religious community, and so on. If I remember correctly, he sees at least six distinct religions: the three core Hindu dogmas, Vaishnavism (followers of Vishnu), Saivites (followers of Siva) and Shaktas (followers of the Goddess); Buddhism, whose dogma and metaphysics is similar to the other three and yet clearly distinct, Jainism, about which much the same can be said, and the Sikh confession.
Philosophical systems criss-cross through this. For instance, Advaita, promoted by Sankaracharya, Vishishtadvaita, taught by Ramanujacharya, and Dvaita, explained by Madhvacharya broadly map on to Saivism, Vaishnavism and Saivism again, although in saying this, we are violating the broad guidance of the Advaita school that an individual may take refuge in any of the five forms of the one God, Ganapati, Vishnu, Siva, Shakti and Surya; we are also about to jump into hopeless confusion when we consider the overlaps between this pre-modern Hindu dogma and Buddhism.
Moral of the story: Don't even try getting involved.