Master's and PhD at different Japanese universities

I have just finished a BSc degree in maths at ETH Zurich and I am now planning to continue with an MSc and a PhD in Japan.

My question is this: is it possible to do an MSc at one university and then do the PhD at a different university?

In Europe this is not only socially acceptable but nothing unusual. But as far as I know, Japan is different; how different I don't know. I would very much like to apply to one university for the MSc and then relocate to somewhere else for my PhD. But I'm rather worried that this may mean that the professor supervising my MSc will lose face. And of course, I can't just ask them directly because they might not tell me the truth out of politeness.