I would caution against any attempt to particularize this to Vijayanagara, but it's broadly correct. In Europe, the primary academic form was scholasticism, a primary component of which is the disputation, where the truth is discovered through arguing on points.

This spread from religious studies to academia and then back to religion numerous times over the middle ages. Most famously, I think, in a religious context, were the debates between Jews and Christians over the rightness of Christianity.

I doubt Vijayanagara had the style imported from Europe, even via the Muslim world. Hinduism had a long history of religious debate prior to the entrance of the Muslim world. We're aware of debates between Islamic Scholars on Hinduism and its place in the Islamic world, and that there were disputations between Brahmins and learned Muslim religious leaders in Delhi both before and after the Mughal conquests is a matter of record.

The Kings of Vijayanagar seem to have embraced a vaguely pluralistic approach to ruling their people. The Hindu sects they adhered to were patronized along with other local practices, and their ongoing war with the Islamic sultanates to the north rarely took on the tinge of religious persecution, even if it was dressed as such. Mosques were common along the coast, though I don't know if there were any in Vijayanagar itself. Religious debate may have been as common in the south of India as it was in the North, as that was generally the preferred method of persuasion for most of the Early Modern Era.

Enthralling? Bear in mind that we live in a very different world to Early Modern India. There may have been far less secular knowledge among the people, but religious knowledge was much more common, especially among the educated populace of the cities. A (probably poor) comparison could be made to Presidential debates.

As to whether or not there were other debates beyond religious matters... as I've hinted, in both Christendom and the Islamic World 'proper' (loosely defining that as the modern Arab world and the Persian lowlands) there were debates going on all the time, some of them public and others long distance. Similar practices were common in China under the Yuan, Song and Ming.

India is a bit of a black box for me. There were, in Vedic times (or at least, in the times when the Vedas were written down), moralistic/legal debates over war and peace, and duty and family, the traces of which can be seen in the Bhagavad Vita. Several religions sprang from debate within pre-Hinduism Hinduism, not all of which were spiritual in nature. Buddhism and Jainism spring to mind.

To the best of my limited knowledge, however, by the early modern era India had fallen behind Europe, China and the Islamic world. The wars that had allowed the Muslims to march in and establish themselves had never really stopped, and the Muslims who ruled most of the subcontinent quickly began their own round of war.

The lands that were Vijayanagar's heartland had never really been on the cutting edge of Indian society. Their final unification of the south changed that, though, and there was something of an intellectual Renaissance in the coastal cities. But...

The primary literary output of the Empire was religious and political in nature. There was incredibly important work done in mathematics and astronomy in Kerala. I don't think there would have been much open public debate on these subjects.

Sorry, I know that's a lot of words to say some things which may not have been helpful and then add in 'I don't know', but it's the best I can manage.