I can speak only from what I've seen, having moved from a History department in Canada to doing Economic History in the UK. The split seems to be real, and if it's being slightly exaggerated, it's not fundamentally incorrect. In North America, there is quite a substantial schism between qualitative and quantitative historians, in terms of their departments, methods, journals, and so on. In the UK and elsewhere, this is still noticeable, but more of a continuum than a clear divide. The split is becoming more pronounced in the UK due to the outsized influence the US academic job market has on global academia, which is a bit sad to see. It looks from over here like the boundaries might be softening a bit in North America, though the debates over things like the History of Capitalism literature shows just how much the whole issue is haunted by the ghosts of old political and methodological disputes...

Many different things are packaged into Turchin's perspective, but the most important of them is mathematics. (More specifically, the acceptance or rejection of mathematically-specified models, and the testing of those models using statistical methods.) History departments tend to range anywhere from vaguely agnostic to openly hostile towards these sorts of methods. Very few historians and even fewer departments are primarily quantitative in their orientation. Certainly they do not tend to include training in the sorts of "empirical" analysis done in economic-history-as-practiced-in-econ-departments. Indeed, even the meaning of the word "empirical" in practice is different, which is why I put it in scare quotes.

Turchin is after something larger than just economics - he is looking to advance an entire field of social science whose research agenda focuses on the development, refinement and testing of mathematically specified models. This itself isn't so distinct from where Economics (and to a lesser extent Political Science and Sociology) are ending up, as there has been considerable drift away from specifically economic theories and topics and and towards a Freakonomics-style (or should I say Gary Becker style?) "Economics as the quantitative analysis of anything and everything". The relatively broad topical boundaries is becoming reminiscent of history departments, where almost anything could be a valid object of study, but the use of mathematics to specify models and the use of all-purpose statistical methods for testing them is not.

More specifically, Turchin is interested in really, really big stories about centuries-long historical cycles involving giant demographic shifts and so on. History is to be modelled as a large system, like an ecology. You need mathematical tools to make much progress even formulating such theories, which is why that's the skill set he thinks students will need. He is convinced that, with enough research, this will be a fruitful way to understand, explain, and even predict historical patterns - a true historical science in the physical sciences sense. Very few people do this anywhere, even in outside North America. And it's like Jared Diamond on steroids, so given the sack beatings Diamond's work regularly gets from history-in-history-departments historians, it is unlikely to find many adherents there.

For my taste, I think Turchin goes too far in his advocacy of these sorts of systems models as explanatory. Quantitative skeptics will tell you that with a few degrees of freedom and fitting a bunch of interacting variables, you can explain just about anything with just about anything else, if you squint. Add in some soft-boiled conceptual categories and some "researcher degrees of freedom" and you can end up in tinfoil hat territory. But this is more about the search for totalising theories using few variables than about mathematics per se; one can also use mathematical tools in other ways than trying to be Hari Seldon. But then, he is trying to found an entire branch of social science, so presumably a bit of ambitious overreach is part of the strategy.