No, I did stats as part of economics around then, and it's nothing like modern DS. It overlaps a fair bit, but in practice the classical stats student is bringing a knife to a gunfight.
The practice of working with huge datasets manipulated by computers is valuable enough that you need separate training in it.
I don't know what's in a modern stats degree though, I would assume they try to turn it into DS.
Data science is basically a marketing title given to what would have been a joint CS/statistics degree in the past. Maybe a double major, or maybe a major in one and an extensive minor in another. And it's mostly taught by people with a background in CS or statistics.
Like with most other academic fields, there is no clear separation between data science and neighboring fields. Its existence as a field tells more about the organization of undergraduate education in the average university than about the field itself.
The Finnish term for CS translates as "data processing science" or "information processing science". When I was undergrad ~25 years ago, people in the statistics department were arguing that it would have been a more appropriate name for statistics, but CS took it first. The data science perspective was already mainstream back then, as the people in statistics were concerned. But statistics education was still mostly about introductory classes of classical statistics offered to people in other fields.
The practice of working with huge datasets manipulated by computers is valuable enough that you need separate training in it.
I don't know what's in a modern stats degree though, I would assume they try to turn it into DS.