"India" as a vague regional term can refer to the Republic of India as well as Pakistan, Bangladesh and occasionally Sri Lanka; these countries are all linked culturally, linguistically (the Dravidian languages borrow lexically from Sanskrit) and geographically. Whilst the subcontinent had been unified by previous empires, was there a notion of "India" like there was a notion of "Europe", and did that definition match today's definition?
/u/Historyguy81 : To them, "India" just meant the area around the Indus river, and later for the region beyond it to the east as well. In Herodotus' time, ~400s BC, the Indus river and the vicinity was the extent of that part of the known world. He describes a diversity of peoples around that area, and says that Asia in inhabited, but that nobody (in his society, remember) really knows what's beyond the border region of the Indus.
/u/xiongnu1987 : The term 'desi' refers to South Asians without distinguishing between country, and is a word they themselves use. It can also refer to native/traditional things (like the term "desi" itself, haha.) This term acknowledges shared history and culture without branding people, who might feel very strongly about Pakistan's own history and issues, as "Indian..." and vice versa. Some of these issues are the necessity/"rightness" of the creation of Pakistan itself, and which country, if any, can/should claim Kashmir.
OP: What do you mean by "notion of 'Europe'?" Europe as a region, or some idea of feeling "European" because of common cultural traits and history, or something else? If you talk about India and mean the country today in our modern notion, then no, that is recent.
For a long time, to people in China and the West as well as Middle-East that entire general area was considered India which is why I find it ridiculous when a Pakistani person gets worked up at being identified as Indian..