Using my university email address as correspondence address to apply for PhD positions in the same university I was just about to send my first PhD application to University XXX. I am currently finishing my master's thesis at University XXX. Would it be incorrect to use my university address to apply for PhD positions?.
Could it be taken as unrightful use of the "university name"?.
I don't see any possible problem with using your current university email address in your application.

The university issued you this  address with the expectation that you would use it to communicate with people inside and outside the university. I can't imagine any way this would be considered "not rightful".