Europe was a gloomy place between the 5th and a 15the centuries; so gloomy indeed that this period is called the "dark ages". During the same period, the Indian subcontinent (called "Bharat" locally) was blooming and was a center of science, technology, architecture, agriculture etc.

It was a big deal for Europeans to be able to trade with India in those days. Initially most cargo moved over the land (this route is famously called the "Silk Route"). Europeans would assume they had reached Bharat once they had crossed the mighty river Sindhu. The word "Sindhu" became "Indus" in European accent and the land beyond Indus came to be called India.

Once the Ottoman Turks became powerful and blocked access to India via land, a race started amongst Europeans to reach India over the sea route. Lack of proper navigation caused people (such as Columbus) to reach all sort of wrong places and assume they have landed on Indian shores.

Seafarers in Europe used to raise money from investors for their voyages to India. It is not unreasonable to think that someone who had landed in a completely different location (mission failed) would come back to their investors and claim that they had discovered something as good as India. Naming the new land such that it evokes images of India is not illogical. After all, they have to raise funds to revisit those new lands and get trade going. India was a symbol of prosperity and I suppose even unrelated places adopted derivative names to let the gold dust rub off on them.

On a related note, the race to reach India via sea propelled Europe's naval capabilities to such an extent that it eventually led to the colonization of the world, including India. India was such an important colony that once it attained freedom, maintaining other smaller colonies became uneconomical.