I can't answer when it was first referred to as the subcontinent, but I can tell you it has absolutely nothing to do with continental drift.

It's not called the "subcontinent" because it happens to sit on a separate continental plate that crashed into Asia 40 million years ago. It's called that because of the results of that collision, which are apparent entirely in the topography (and to a lesser extent, vegetation).

If you look at a relief map of the subcontinent, you can see that it is almost entirely walled off from the rest of Asia. There are only a few gaps where people can get through - the Khyber Pass through Afghanistan, a short stretch of pretty inhospitable coastline in southern Baluchistan, and the jungles of Assam in the east. Either there are tall mountains blocking the way, or these is dense tropical jungle (which until recently was also home to headhunting tribes).

This seals off the subcontinent from the rest of Asia, which means that historically, civilization in the subcontinent developed and went its own way without a lot of contact from the outside world. This is reflected in languages, culture, cuisine, shared history, etc. The subcontinent has kind of always been its own world.

Of course, these features are due to plate tectonics and continental drift, but you don't need a theory of plate tectonics to call it the subcontinent. The actual physical features produced by continental drift, in the form of those big mountains and the resultant impediment to human migrations, have been around for tens of millions of years. We look at the fact of the isolation to call it the subcontinent, we don't need to know the geological why of it to make those categorizations.