Nobody owns the rights to words, or the "rights" to names, history, etc, etc.
I can assure you that there would be hundreds (thousands?) of teams called the "Eagles" in the world.
What you can do is design and register trademarks, for specific purposes.
The big issue is of course merchandising.
Back in the 80's & 90's, you already buy NBA gear here. It was all licensed and the trademarks are registered in Australia by the NBA.
BIG business like that often design and register not only the ones currently being used, but variations, and prospective designs. If you do a search there will be various combinations of the scripts "Phoenix", "Suns," and images of suns and basketballs.
Again, that doesn't stop you calling your team the suns. And you COULD register your own trade-mark, including featuring the script "suns" provided that it doesn't LOOK like the NBA variants. Problem is when you try to combine it with an image of a sun or basketball.
I haven't checked the details of the NBA trademarks, but its POSSIBLE they are only for merchandise and clothing in Australia. So you could possibly even get away with playing in the NBL. (That would only be effected by TM's registered for live performances, entertainment, etc.)
But obviously Townsville couldn't sell clothing or merchandise, and even their playing singlets could be a problem.
Fortunately for Sydney, firstly they might have beat the NBA to the punch anyway, but irrespective their TM's logos are distinctive and different. (Even the older Sacramento logo features the script "Sacramento" and a basketball is only a small part.)
There's a probably a good reason why the Sydney logo doesn't just feature a large crown, as there would doubtless be numerous such unoriginal TMs.
Words don't matter, its the total image that makes up a TradeMark.
Suns + Basketball = No Good.