Yes, both because India is an ancient name for the region still broadly coveted by that name and because ancient people in India agreed with the general geographic boundaries while using their own names the region.
The name "India" originated in ancient Greek as a simple translation/transliteration of the ancient Persian province Hindush, which was just the Iranian languages' version of the Sanskrit word Sindhu, the Sanskrit name for the Indus River. So the ultimate origin of "India" was to describe "the region around the Indus."
However, "India" was mostly an abstract concept to the Greeks. It was unimaginably far away and they started using India to mean "everything east of Persia" before any Greeks had actually traveled to the Indus River itself. Once Alexander the Great's conquests broadened Greek understanding of geography a bit their use of the name India got more specific. Here's how "India" (technically using the alternate spelling Indike / Ίνδικη) was defined by Megasthenes, a Greek who actually traveled there around 300 BCE:
India then being four-sided in plan, the side which looks to the Orient and that to the South, the Great Sea (Indian Ocean) compasseth; that towards the Arctic is divided by the mountain chain of Hēmōdus (the Himalayas) from Scythia, inhabited by that tribe of Scythians who are called Sakai; and on the fourth side, turned towards the West, the Indus marks the boundary
Compare that to Chanakaya's Arthastra, written around the same time:
The Brahmaputra River is the eastern boundary of Jambudvipa (India), its western boundary being the mouths of the Indus and its southern boundary being the Indian Ocean
Similar boundaries were described in the Vishnu Purana about 200 years later, Arrians Anabasis of Alexander 200 years after that, and the Chinese travelog The Five Indies in 650 CE. So the idea of a "India," using a variety of names in different languages was widespread in the ancient world. Not only is it a region surrounded by obvious and harsh natural boundaries, but the region within those boundaries had a shared culture. Elements of religion, language, and other traditions were shared across the subcontinent and the surrounding region.
I'm guessing that this question is at least partially inspired by the frequent claim that India was never united before British colonialism. While technically true, the idea that implies to a lot of modern people isn't historically accurate.
Just because India was not a governed by a single state administration didn't mean that all of the different polities in India were completely unique and hostile to one another. Over the course of about 3000 years very few regions in modern India, Pakistan, and Bangladesh were completely independent at all times. Most of the region has conquered and/or been conquered by their neighbors at some point, and each instance further integrated and shared local cultures across larger areas.
Most of modern India and Pakistan actually came under single empires twice in pre-colonial history. In antiquity, the Mauryan Empire controlled all of Pakistan, parts of Iran and Afghanistan, and most India with the exception of heavily forested regions and Tamilakam. Then, in the 17th Century CE, the Mughal Empire spread to control nearly identical borders to the Mauryans for a time.