Do EPSRC-like Centres for Doctoral Training exist outside the UK?
That's very country and scientific domain specific and based on policy of government which may change every election cycle.


e.g. in UK the established UKRI encapsulates research councils STFC,EPSRC,MRC, ... and fulfils UK specific policy.
AFAIK in Germany there are 4 major non-university institutions doing research and PhD training - Max Planck Institute, Leibnitz Institute, Fraunhofer Institute, Helmholtz centres and many others - some of them multidisciplinary.
As I'm currently in Czechia (Czech Rep.) there is Academy of Sciences and it's centres some of them connected to relevant universities with doctoral training program. But most are based on individual labs and mostly individual training with courses available in multiple institutions.


Based on my personal experience in UK, in Czechia and little in Germany I cannot generalize. Thus you should do your own research in your domain of interest, find rather individual lab which may fit your scientific domain in a country you'd like to study.