In the Potterverse, Dumbledore had a Phoenix named Fawkes.
Was Fawkes the only Phoenix in the Wizarding world?
Some evidence for other phoenixes:
In Quidditch Through the Ages, there’s mention of a Quidditch team with a phoenix as their mascot:
Antipodean teams have always thrilled European crowds with their speed and showmanship. Among the best are the Moutohora Macaws (New Zealand), with their famous red, yellow, and blue robes, and their phoenix mascot Sparky.
This is the only other named phoenix in the canon. (Don’t ask me why the mascot isn’t a macaw.)
Ollivander remarks that Harry and Voldemort's wand have tail feathers from the same phoenix (Fawkes). If phoenixes were rare (or Fawkes was the only one), then it would be more remarkable that Harry's wand had any phoenix feather.
Instead, phoenix feathers are described as one of several possible wand cores. Indeed, when Harry first visits Ollivander, he's offered another wand with a phoenix feather core:
“Maple and phoenix feather. Seven inches. Quite whippy. Try —”
Harry tried — but he had hardly raised the wand when it, too, was snatched back by Mr Ollivander.
Clearly Fawkes is not a unique specimen.
Indeed, there are at least two other wands with phoenix tail feathers, which surely cannot come solely from Fawkes.
They're described in Fantastic Beasts and Where to Find Them:
The phoenix gains a XXXX rating not because it is aggressive, but because very few wizards have ever succeeded in domesticating it.
I'm not sure whether you could describe Fawkes as domesticated, but this again tells us that there are other phoenixes. Otherwise, why mention them?