SURAT, India—Founded just six years ago by a 19-year-old entrepreneur from one of India’s poorest states, Oyo Hotels and Homes has grown so fast that it already ranks among the largest hotel chains in the world.

In fact, going by the number of properties it claims in its network, Oyo is by far the biggest, twice the size of No. 2 Wyndham Hotels & Resorts.

But...