are they really all that different?

This article from the GNU Project is helpful in clarifying why, Here's the main gist though:

The term “intellectual property” is at best a catch-all to lump together disparate laws. Nonlawyers who hear one term applied to these various laws tend to assume they are based on a common principle and function similarly.
Nothing could be further from the case. These laws originated separately, evolved differently, cover different activities, have different rules, and raise different public policy issues.

it means making a copy on their own servers?

That's also murky since the dataset the OpenAI trains on is a trade secret, and most likely astronomically huge with not just NYTimes but other publications as well. Microsoft and OpenAI haven't responded, I doubt they didn't already have a plan beforehand (which was most likely what we've already seen, lobbying/compaigning against "AI" in order to become regulators of their own field). The hysteria and disinformation around machine learning and large language models is also part of OpenAI's strategy to muddy the waters.