We can surely do better than two one-word answers!
I'm not a linguist, but I suspect the 'eurocentric' rub is more in the notion that languages are today a 'unified pan-national encompassing element of identity'. Look at any number of African countries, for example - South Africa has eleven official languages and sings its national anthem in five. English is only a kind of lingua franca, and speaking English is not part of South African identity - in fact, I've met plenty of people out there who can barely understand it.
Bah, really I'm just kinda lazy right now, but that's really not it either. Basically the paradigm upon which people think about languages is wrong. Well, not wrong; it works to an extent in the modern day, and it works historically if you're just talking about the literate elite class of any given region, but otherwise it is wrong.
So here's how you have to think about it:
1) If you know anything about Germany, think about Germany pre-1950. Essentially there were two "Germans" spoken in any given region. There was
Hochdeutsch which was the proper, formal German that was spoken on the radio and on tv, in formal political addresses, and in writing. If you took German as a foreign language, this was the German you were taught. Then there was the local dialect. Every village had a local dialect distinct to the village through which locals communicated with one another, and which you almost never taught to outsiders. This dialect was related to
Hochdeutsch, sure, but it was decidedly its own thing, linguistically speaking. And although the village dialects were close enough that someone from, say München could communicate with someone from Nürnberg, once you get further afield to into Styria or Sachsen those village dialects no longer become mutually intelligible.
2) The above is what is called a linguistic continuum. To better envision what this is, consider this second, more relatable example. So let's say today, in the year 2017, you start on the west coast of Sicily. You find some random villager (villager A) walking around, and task him with going to the next town over and talking with somebody over there to see if the version of Sicilian villager A uses is mutually intelligible with the Sicilian spoken by a villager B. Villager A walks to the next town, talks to villager B, and finds that, yes, Village A Sicilian and Village B Sicilian are mutually intelligible. Villager B goes to Village C and finds that B and C are mutually intelligible. Now, you can continue this daisy chain from Western Sicily, across the island, across the Strait of Messina, up the boot, through Liguria, through the South of France, over the Pyrenees, through Catalunya, up through Castille, and end at A Coruña, and at no point would that daisy chain be broken. That is- at no point would any given villager be unable to speak with someone from the next village over. Even though in the macro-scale, the Galician of A Coruña is demonstrably, self-evidently different from the Catalunyan of Barcelona, from the Italian of Florence, and from the Sicilian of Messina; at the micro-scale, you would never be able to point out where exactly the Galician ends and the Portuguese or Castilian begins.
That is a linguistic continuum: a continuous geographical stretch of closely related and interlinked dialects that form a band of gradual change. Examples would include the Sicily-Italy-France-Spain-Portugal/Galicia continuum I just mentioned above; the Northern/Western Germanic band of the various German Dialects of Switzerland, Austria, and Germany, plus Plattdeutsch, Nederland; and the North Germanic band of Danish-Swedish-Norwegian, etc. . These continua have faded rather dramatically over the last 200 years due to a) the proliferation of a written grammatic standard, and national, standardized compulsory language education, b) the rise of the nation-states, with national governments legislating the destruction of any and all local and regional dialects in pursuit of a more coherent and consistent national identity, and c) the rapid technological improvements in travel and communication (particularly tv and radio) necessitating and hastening the increased sublimation of local and regional dialects in favor of a more broadly intelligible standard dialect/accent.
However, before those trends started to develop, namely pre-c. 19th century, the continuum was the norm, and it is arguable that the whole of Europe, if not indeed the whole of Eurasia might have been one enormous linguistic continuum. It's important to remember that "language" is only a discrete, definite phenomenon inasmuch as we make it so for the ease of communication (viz: to be able to say "English" rather than "The assorted related dialects spoken in the US, UK, Ireland, Canada, Jamaica, Australia, India, New Zealand, and the various Pidgins spoken in West Africa and the Caribbean") and categorization (because human nature is to make sense of an inherently nonsensical, chaotic world), just as the delineation between species is in biological terms can often be a bit arbitrary and is, at the end of the day, something we apply post-facto for categorization purposes, rather than something that exists inherently in the system. There's no such thing as "Early Modern English" or "Classical Latin"; these are catch-all terms to refer to any hundreds if not thousands of variants and dialects and regionalisms that may have existed over however long is useful to us for the purposes of neat and immediately understandable geographic and historical categorizations. The same can be applied to Language Families. In the abstract Slavic and Germanic are obviously different families, but that is because the standardized, modern descendants that have come to represent these families grew up in geographically and linguistically very distant contexts. If we went to, say 1340, pulled a Brothers Grimm, and simply started documenting the forms of language we were hearing along the Oder, the sorts of dialects we would be hearing would probably not fit quite so neatly into our Linguistic Family archetypes.
We don't think about think these things because they simply aren't the reality that we inhabit today, and it's very easy to forget that linguistic reconstructions are themselves abstractions representing any hundreds upon thousands of different local realities. It certainly also doesn't help that languages, particularly regional and non-aristocratic languages only really started being documented with any kind of academic rigor about 200 years ago.