Phar Lap's strapper, Tommy Woodcock was no different to any other stable hand, they all nickname horses.
Bobby, Billy, Curley, Freddy, Jacko etc, all names that are commomly used, no stable workers ever refer to the horse by their registered name, it's more convenient to shorten the name to something more personal like Bobby, plus it's good security at trackwork and around the stables, as the only ones who know the nicknames are the stable connections.
Unlike 'Bobby' the names 'Big Red' and 'Grey Ghost' were not stable nicknames but were descriptive titles created by the press to add excitement to those horse's status.
For what it's worth, Phar Lap was also, and still is referred to as 'Big Red' here in Australia.
I don't see anything weird about the name 'Bobby', if Woodcock had nicknamed him 'Rumpelstiltskin' then, Yes, I'd agree it was a weird nickname.