In my Middle Ages class today my professor said this, saying that historians in the 19th century used 12th century notions of what Christianity should be to define other religions and therefore "invented" Hinduism as a religion. Is this accurate?
It isn't entirely accurate, but a simplification of the history. Hinduism is a diverse set of loosely related religions originating in India. Nearly all forms of Hinduism focus on the Vedas, Upanishads, and to a lesser extent, the Puranas. These traditions all diverge in their interpretation of these books and their application. For instance, not all Puranas and Upanishads are universally accepted or applied to all forms and many have supplementary texts and rituals.
Most forms of Hinduism have their roots in the Brahminical Orthodoxy, i.e. the Vedic and post-Vedic religions centred around a hereditary priesthood. The development of Vishnu and Shiva, arguably the two most popular gods in Hinduism today, began in the late Vedic period and continued to develop until they had all but replaced classical Hindu gods in the 1-3rd centuries CE. While there was a certain unification in these traditions seperate from Buddhism and Jainism, rarely were the forms of worship considered the same. It was unlikely for one to worship both Shiva and Vishnu as equals and each had their own traditions.
The major unifying factor between these traditions, aside from the shared holy texts and priest tradition, was the Vedic philosophical thought. Perhaps the most successful of these were Yoga and Vedanta, specifically the Advaita version. Both traditions were wildly popular with the Vedic traditions and served as a unifying point.
During Islamic rule, the traditions of India were periodically suppressed by the Islamic ruling class. To them, especially the early ones, there was little difference between the traditions. That said, some Islamic rulers, even in the famously iconoclastic Delhi Sultanate, did see a unifying force between Hinduism and Islam, especially Vaishnavism. The point though is that the Vedic traditions were grouped together during Islamic rule for the most part and unified in opposition to repression and later as a social and economic force in the Mughal court.
The colonial powers DID help codify Hinduism into a single unit, the treatment and common idea of Hinduism as a polytheistic religion by western conception. And to some extent, it became that via the influence of neo-Vedantans, and later Gandhi, who attempted to spiritually unify all Indian traditions to create a sort of Indian national religion.
As a follow up question: what's the take of indian/hindu scholars on this (alleged) western definition?