Sometimes questions are very short and answers very long.
We all like coffee, but, would you give coffee to your dog?
According to dictionary.com, one of the definitions of the word drug is:
a chemical substance used in the treatment, cure, prevention, or diagnosis of disease or used to otherwise enhance physical or mental well-being
As caffeine, which coffee contains, is an active ingredient in some medicines (such as Anadin) it certainly fits that definition so it would be correct to say that coffee contains a drug (putting aside decaf). If you take the first sentence of the Wikipedia page that describes drug:
A drug (/drɑːɡ/) is any substance that causes a change in an organism's physiology or psychology when consumed.
Caffeine also fits that definition as there's a body of evidence that supports that description when talking about caffeine.
Would you give coffee to your dog?
Absolutely not. There is a page on petmd.com titled Caffeine and Pets: Safety Tips and Considerations which states:
It turns out our pets react in much the same way we do. Caffeine makes them restless. They get jittery and their hearts start to race.
That doesn't sound.... ideal! It turns out that it's not, and the next part of the page makes it pretty clear why:
But because our pets weigh so much less than we do, it only takes a relatively small amount of caffeine to cause a big problem, potentially leading to expensive hospitalization or even death.
Further on in the article it's explained that 23-27mg caffeine per lb of body weight can lead to cardio-toxicity. For a 3.5kg cat, such as mine, that equates to between 177mg and 207mg of caffeine. Based on the table in this answer, that puts a double espresso or a 200mg caffeine tablet squarely in that range.
That's absolutely not to say that giving your dog, or other pet, coffee (even decaf!) is something you should consider doing. The petmd page states it much better than I ever could (my emphasis):
“Cats and dogs should not ingest any caffeine,” says Dr. Elisa Mazzaferro, adjunct associate clinical professor of emergency-critical care at Cornell University College of Veterinary Medicine in Ithaca, New York.