In general terms in most markets there is a BX2380, and a BX23S. The BX23S is just a BX2380 with a backhoe on, so in a sense the question "can a BX2380 take a backhoe" is a little complex, because the answer is that if it had a backhoe it would be a BX23S.
The thing that's different about the BX series is that Kubota have decided the tractor is too small/light to just have a backhoe bolted to it, so when you put a backhoe on there's also a full subframe that goes on to reinforce the chassis. In concept you could put that subframe and the backhoe on a BX2380, but it'd be much cheaper to just buy the BX23S originally.
As for the BX231, I'm not familiar with the model. I've looked online, it looks a lot like a standard BX, but perhaps with an EU specific set of emissions controls on the engine. It doesn't have the BX80 series flat face couplers, and there's no mention of a backhoe being available. Which perhaps leads to the question of whether a US-spec BX backhoe and subframe would attach, or whether there are differences that would make it hard. Either way it'd probably be expensive.
EDIT: in terms of the next model up, I've looked through all the models listed on the Kubota EU website, and the brochures make no mention of backhoe. I wonder whether that's because there isn't a market for it, or perhaps some sort of health and safety regulations that mean they can't be sold. Certainly the tractor models look smaller than rest of world, but have a lot of quite premium features on smaller machines than we'd see here in NZ for example. So I see bi-speed turn on B class machines, which we definitely don't see, and cabs on much smaller machines than we tend to see them on here. I hadn't realised how different EU spec models are.
I wonder whether you can buy a tractor outside the EU and import it - does anyone actually compliance check private imports? Particularly if you can bring it over a road border I'd presume Customs have better things to do with their time than understand whether it's an EU spec model.