> Since when have computers been “basically built for copying”
Digital computers (i.e. Shannon's Information Theory) was a solution for the S/N getting worse every time the signal was amplified. Shannon's solution of digital circuits allowed information to be copied without the limit. An important consequence of this was the marginal cost for copying information falling very close to zero. Or, stated in economic terms, information ws no longer scarce.
> computer
You're using a very narrow definition of "computer". Colloquial use of the word obviously includes a variety of technologies and concepts - such as Shannon's digital circuits - not just model of computation (Turing or otherwise).
> modern computers are effective communication device
You don't even need to consider {,inter}networking; the modern devices commonly called "computers" spend most of their time managing data. Very little time is spent doing any actual computation. I recommend this[1] lecture by Feynman, where he explains how what we call "computers" are are really closer to a filing system that stores and copies data (including the store program and most of the mechanics of computation).
[1] https://www.youtube.com/watch?v=EKWGGDXe5MA