. . . or was he never even given a chance specifically because of who he belonged to?
The Romanov's dog Joy was taken in and cared for by a guard, so I don't think there was any particular animus against the dogs because of their ownership. I think the best guess is that the dogs lived or died based on which ones did or did not get in the way during the run-up to and the wrap-up from a massacre.
To people who were murdering eleven people that night -- including children -- and trying to keep it under wraps, it seems a little unrealistic to expect them to have paid much attention to dogs, one way or the other.
I think rgt9w brings up an interesting point. The Bolshevik takeover and ensuing civil war unleashed vast forces of pent up resentment, anti-semitism, xenophobia. And the relatively sudden release of those forces resulted in chaotic horrors of all sorts across Russia on both the White and the Red sides. (It's worth reading Figes'
A People's Tragedy to get a sense of just how universally violent and even sadistic the period became.)
I suspect the wholesale killing of Borzois based on association and the killing of the Romanov dogs tracked a distinction worth noting in the treatment of their masters. Everything about the way the Romanovs' captivity and murders was managed reflected cold calculation for a specific political end. Many members of the aristocracy were killed by mobs in the throes of an inchoate and unreasoning fury.
If anyone was motivated to kill the Romanovs' dogs to make a political statement, the dogs would have been killed while the Romanovs were alive. You can't make a point to a dead person.
