"What does this "non-relational" aspect mean, that Dasein does not
relate to its own death?"
Possibly one could say that. It means it is not relative, but absolute (more precisely: not relational, but that which can't be "outstripped" [that's important insofar as one doesn't want to get the philosophic & theological context of the word "absolute" mixed in here, but it is easier, and, I believe, not misleading, to understand it by saying it this way]). Someone with a cold is relatively unhealthy. Death is ultimate. One could say, developing the example, health is something that in each case is known by each Dasein, from the essence of health, the cut is relative to one's knowledge of the essence of health. We don't have knowledge of the essence of death according to the work with the name Heidegger. Or, said with greater exactness, knowledge of our own death.
It goes without saying that Death is not here understood under the biological conception. As one will have already understood.
'Apres coup' (after stroke or thought): After thinking this over, I want to thank the questioner for guiding us in this direction. Something more urgent comes out of this, which is that Da-sein is related (cf. the Davos debate) primordially to Truth (there is a question as to whether it continues to be appropriate to term this basic trait truth, or conceive it so, viz. the remarks in On Time and Being [Zur Sache des Denkens]), but if it is not related to its-own-death, this shows death as a counter to truth and falsehood. One is then drawn to consider how death stands alongside nichts nichtet, the Nothing (as a specific modification of the orginary Greek conception of Chaos, the limitless or unmeasured gulf).
This answer must not be taken for an explication of the text. In fact, as "jobemark" rightly pointed out, we have not even checked the original text here, against the translator's interpretation (as the text called Heidegger understands all translations). It is an answer to the question highlighted above. The simplest and perhaps most-telling explication would involve adducing something from Tolstoy's Death of Ivan Ilyich. Keeping in line with what "idiot an" says: "no-one can die my death for me".
‘The syllogism he had learnt from Kiesewetter's Logic: "Caius is a
man, men are mortal, therefore Caius is mortal," had always seemed to
him correct as applied to Caius, but certainly not as applied to
himself. That Caius—man in the abstract—was mortal, was perfectly
correct, but he was not Caius, not an abstract man, but a creature
quite, quite separate from all others.’
The Death of Ivan Ilyich by Leo Tolstoy