What is the nature of the "assistant professor" position in India?
Indian universities are not very homogeneous, but it may be helpful to divide them into two categories:

(1) National institutes/universities:The top-tier universities with highly competitive admissions and recruitment processes. A commonly stated figure is that 1% of applicants make it to these institutes, though I think it may now be closer to 10%, with more institutes opening in recent years. In this category, an Assistant Professor must hold a PhD and have 3 years post-doctoral experience.

(2) Other universities: One requires a Master degree and must qualify a national-level entrance exam to become Assistant Professor. Yes, one may continue in this post without a doctorate, though in practice this is rare. Full professors (and perhaps associate professors too- not sure on this) require PhD.

The second category might appear strange from a US perspective. It may help to recognize that these universities tend to place pedagogy over research. This is an outcome of having a large population and limited employment prospects without a college degree.