On a real-world level, it makes sense that the episode would acknowledge the New Eighth Doctor Adventures and Big Finish rather than, say, the novels, since they're characters played by actors that Paul McGann has actually met and worked with for a number of years. Whereas it's entirely possible he's never even heard of, say, Destrii or Fitz. He and India Fisher are actually the single longest running TARDIS team by continuous years! It's only right that it's Fisher, Conrad Westmaas, Sheridan Smithet al. that get acknowledged when he finally appeared in the TV series again.
In-narrative, there's even a possible explanation for why he referenced those companions rather than his companions from the novels and comics or his later audio companions: he mentioned only the companions that (as far as he believes) are already dead. There were other Big Finish companions, after all, that he failed to mention in the episode.
It is noted in Soundtrack Dissonance that the triumphant music that plays after the regeneration seems odd when the War Doctor's birth is not seen as a triumph to his future selves, however it may in truth be foreshadowing that in fact the War Doctor is not as terrible as his counterparts believed, with his greatest sin having never truly happened.
In the Tenth Doctor comic story, The Forgotten, he has a flashback to his life as the Eighth Doctor and reveals that this incarnation died alone. The regeneration story not only confirms this to be true, but that the Eighth Doctor actually demanded that he be alone when he regenerated.
When he regenerated, the War Doctor was a young man. Given how slowly Time Lords age even in a single regeneration, exactly how long was he fighting to end up as the old man that we meet in "The Day of the Doctor"?
