No, i dont think there was usage of any identity called european before the modern 21st century or the creation of the european union even now european terminology is used to denote the specific ethnicitie of northern and western european descendents in america and not others such as spaniards, portugese etc who are grouped as latinos in america. The terminiology such as ''indian'' has been attested far earlier as early as the achaemenid empire, the hindu arabic numeral system for instance was used to describe numerals from india, the people from indian subcontinent were always denoted as indian and not gujarati, sindhi, bihari etc, the arab chroniclers give examples of indian traders and their eating etiquittes (eating in separate plates using right hand etc), this is never the case with europe in the first place while india was exposed to large eastern and western communities since the very beginning. The hadis or saying of the prophet muhammad pbuh also used the terms such as indian incense for the sandalwood or tamarind/tamar hindi for tamarind/imli. The is no such case for europe.
Hence there has been distinct cultural similarities which has lead to cultural and geopolitical usage of the term indians etc
There is also attested separate nationalities of indian in south east asia for example kalinga (orissa, a state near bengal), tamils, dravida (probably andhra pradesh) though but that is not the case with western historians.
regards
The usage of the a single term for South Asia vs Europe has more to do with the lack of a detailed understanding of the complexity of the region. You can't used flawed/misinterpreted descriptions to justify a flawed understanding of the region.
Was South Asia ruled through conquest and force by various empires? Yes
Was it, at various points in history, an amalgamation of Princely States, Kingdoms, empires, vassal states etc? Yes
Did it have distinct geographical regions with unique cultures, languages (and after Islam) religious identities? Yes.
So on what empirical basis can you claim it had an identity as a single nation-State any more than the Greek, Roman, Mongolian, Persian, Arab, British Empires can claim to be a 'single nation-state'?
Even the geopolitical entity of Europe as in the EU is constituted by multiple nations.
And even there the situation isn't analogous, since the EU is a Union of nations endorsed by the various democratic processes within the nation-States i.e. the people of the Nation-States the comprise the EU.
South Asia, even when large parts of it were unified under a single ruler, has only had 'unity' as a result of forced conquest. These 'unions' are no different from the Mongol, Persian, Arab or other empires that at some point included parts of South Asia.
The one time when the 'people' were actually asked to voice their opinion, it resulted in the Nation-State's of Pakistan and India.