I am now studying at XXX University as an exchange student. Is this my "exchanging experience" or my "exchange experience"? Am I actually an "exchanging student"?
You are not an “exchanging” student, although at one time you would have been an “exchanged” student.
Originally “exchange students” were students who participated in an actual exchange of students between two institutions or sponsoring organizations, with an equal number of students moving in each direction, as in this 1923 account of the program of the Scandinavian-American Foundation:
After the 1960s, this sort of reciprocal program, funded by the institutions or organizations, evolved into less strictly balanced programs, funded largely by the participants themselves(or their parents). By and large, these programs are now called “study abroad" or “cultural enrichment” programs or something of the sort rather than “exchanges”; but the old label “exchange student" has stuck.
You can call your experience an “exchange experience” if you like, and everyone will know what you mean: not your experience of an actual exchange, but your experience as an exchange student.
You are an exchange student having an exchange experience.