How did the term hind (as in hinduism/hindu) turned into a religious term from a geographic term?

The word "Hindu" originally referred to the Indus river. The Sanskrit name for the river is "Sindhu", attested as far back as the Rig Veda from around 1,800-1,500 BC. In the early first millennium BC, the "s" sound changed to "h" in Proto-Iranian, and "Sindhu" appears as "Hindu" in the Zend Avesta. Around the time of Darius (~500 BC), the meaning expanded more generally to "the land around the Indus" and a few decades later in the time of Xerxes it expanded to "the land around the Indus plus everything further east". In modern terms, this would correspond to northern India and Pakistan.

"Hindu" and "Hindi" were also used to refer to the people living in these lands. They are different forms of the same word. "Hindu" is Old Persian while later on there is a sound change to "Hindi". This usage of Hindu/Hindi remained current through the first millennium AD.

Hints of a religious meaning appear after the Muslim invasions, beginning around 1,000 AD. The earliest known reference is from some Vaishnav texts dating to the 13th century, and from the Raj Tarangini (early 15th century Sanskrit chronicle of the kings of Kashmir). It’s not explicitly religious at this point, but religious by implication in the sense "natives of this region as opposed to those Muslim invaders". But soon after, it becomes clearly religious, for example in the writings of Vidyapati, who refers to "Hindu dharma" and "Turko dharma" (Islam). Vidyapati was a 15th century poet from modern Bihar, who wrote in both his native Maithili as well as Sanskrit.

The earliest European usage of "Hindu" as a religion is from Jesuits in the Portuguese colony of Goa, around the mid 17th century. They used "Hindu" to refer generically to native Indian religions, including what we call Hinduism, Buddhism, Sikhism and Jainism. But not Muslims, who were recognized as a separate non-native religion. It was only in the mid 19th century that European scholars began to distinguish between various "native Indian" religions, and separate Hindus from Buddhists, Sikhs, Jains. Even then, such distinctions were scholarly and did not trickle down to law. The British continued to group all of them collectively as "Hindus" for legal purposes until independence in 1947.

So "Hindu" is an Old Persian word, derived from Sanskrit, that originally referred to the river and the land around it. In Old Persian, the same word also applies to the people of this land, while modern Farsi grammar changes it to "Hindi" when you refer to the people. The addition of a religious identity to these people dates to the Muslim invasions, soon after 1,000 AD. Further refinement of the meaning to distinguish Hindus from other dharmic religions was done by Europeans in the 19th century.

These are all foreign terms. In India, people either referred to their own religion by sect (Shaiv, Vaishnav, Shakti, etc.) or more generally as followers of Sanatana dharma, meaning the eternal law. For Indians of the time, Buddhists and Jains were also just another sect - Baudh or Jain. It was Europeans who classified them as separate religions.