Why are English words with Greek roots so often prefaced with “it comes from the Greek word …”?

I can’t replicate your claim. Google gives 8 million hits for “comes from the Greek”, and 9 million for “comes from the Latin”.

Of course, you’ll see disproportionate numbers of “comes from the Greek” compared to “from the Norse”, “from the Germanic”, “from the Old English”, or “from the French”, because all those languages contributed to English as part of its organic development. Greek contributed to English as a learnèd source of words, and Greek loans in English are usually pretty noticeable; so they are singled out as learnèd loans.

But I’d say the same happens with Latin loans. Maybe Greek more than Latin in some contexts, because Latin is the default source of learnèd loans, and Greek is not.