This induced a little painful wincing. Chemical engineering relies heavily on detailed knowledge of fluid mechanics, because industrial chemical processes generally take place in liquid or gas phase, often under conditions of high pressure and temperature, and so you do indeed need the same kind of general approach that the other listed fields of engineering apply. They rely on mathematical models, to ensure that you don't get explosions (ideally anyway), etc. They need to be able to calculate things like the pressure at the bottom of an oil storage tank and so on. Look at any textbook on fluid mechanics for chemical engineers, it's the same kind of thing.
The author may be thinking of something like organic chemists who work in tamer lab settings, devising new reactions at benchtop scale, but organic chemists generally aren't going to be able to scale up a promising benchtop process without the input of a chemical engineer who has a detailed knowledge of fluid mechanics. Nobody refers to such organic chemists as 'engineers' in my experience, I think the more generally accepted terms are 'researcher' or 'scientist'.