In all honesty it has been a while since I did the Azura questline. But I do remember that the story was basically that this mage wanted to put his soul into Azura's star in order to make himself immortal, yes?

So wouldn't Azura's star work as a phylactery, making him some sort of lich?

Nope, because his body was decomposed and only his soul existed within the Star. Thus, he was technically 'immortal' as long as his soul existed within the Star.

Apart from that, we don't know whether it is even possible to "store" your soul in this way without damaging the artifact (which Malyn did, making the Star require more and more souls in order to stay intact).