Terminology involved makes everything lie a bit.
All the "immortality" and so complicate things a bit. We perceive life as life of our individual bodies. Thus if our body is immortal, that means that it is persistent, without us bothering about our individual cells.
But, there isn't any distinction like that in single cell organisms where cells themselves are taken as "individuals". When cell divide, there isn't usually any distinction of which one is "mother" and which one is "daughter". Both new cells are usually the same. So either it may seem that cells are immortal because they are not dying by old age, or that we define new cells as new individuals and thus the old one died to give birth of two new ones.
But we could take certain aspect of this viewpoint on multicellular organisms as well. Our bodies are defacto created by very old cell-line (or in sexually engaged organism, two cell lines) and then this cell-line created vehicle for itself. From this viewpoint, which is not very different from viewpoint taken when speaking about these laboratory cell-lines, whole life is single cell-line. And from this point of view, life itself is immortal.
So the whole problem is that our point of view, on which we define immortality and individuals, is made only for special types of organism, as has no sense when speaking about single cell organisms. Immortality of cancer cells is thus only buzzword.