Your supervisor assigned you the task of reviewing a colleague's work, so doing that review is your work.
It is also entirely reasonable to send the review comments by email instead of verbally, as it only helps everyone involved to check if all the review comments have been resolved. Moreover, if either you or the colleague gets pulled into another task and/or this task is put on hold, it helps to avoid repeating the work the next time it is taken up.
I would advise against bringing this up to the supervisor, and certainly not using the "not my job" tone being used here, especially if you are a senior staff.
Since you ask, in a formal or semi-formal review, it is more or less standard for the reviewer to write the review comments, and the implementer to respond to the review comments, explaining how it has been resolved or why it cannot be resolved.
The only case I can think of where the implementer takes notes of review comments as the reviewer gives his opinion is when the "review" just happens by chance. For example, the reviewer happened to look at the code while working on his own task, and found something "interesting" that needs to change. In that case, the reviewer might just call and say, "Hey, I found this thing in this code that may need some rework, mind taking a look at it when you have some time?"