Yes, usually the ESR should be also registered as student. And yes, s/he is subject to tuition fees. HOWEVER, the good news is that (at least in our ITN project, based in several countries incl. UK), the tuition fee is paid by the project money, but they are covered from the institutional expenses, NOT from your salary. In other words, even if the ITN hasn't negotiated an exemption from tuition wih the uni, the tuition will be covered for you at no extra cost for you.