It was likely because of his close relationship with Fawkes.
From the evidence in the books, the most likely reason that Dumbledore has a phoenix Patronus, and that a phoenix is representative of Dumbledore, is that he clearly has a close relationship with his phoenix, Fawkes. He tells Harry that phoenixes make very faithful pets.
“He’s really very handsome most of the time: wonderful red and gold plumage. Fascinating creatures, phoenixes. They can carry immensely heavy loads, their tears have healing powers and they make highly faithful pets.”
- Harry Potter and the Chamber of Secrets, Chapter 12 (The Polyjuice Potion)
He also tells Harry that only by showing him extreme loyalty would Harry have been able to call Fawkes to his aid, further showing the connection between Fawkes and Dumbledore.
“First of all, Harry, I want to thank you,’ said Dumbledore, eyes twinkling again. ‘You must have shown me real loyalty down in the Chamber. Nothing but that could have called Fawkes to you.”
- Harry Potter and the Chamber of Secrets, Chapter 18 (Dobby’s Reward)
Fawkes helped Dumbledore in various ways, like when he needed a lookout who could warn him.
“Dumbledore was now stroking Fawkes’s plumed golden head with one finger. The phoenix awoke immediately. He stretched his beautiful head high and observed Dumbledore through bright, dark eyes.
‘We will need,’ Dumbledore said very quietly to the bird, ‘a warning.’
There was a flash of fire and the phoenix had gone.”
- Harry Potter and the Order of the Phoenix, Chapter 22 (St. Mungo’s Hospital for Magical Maladies and Injuries)
In addition, having a phoenix as a pet is very rare - it’s not like having an owl. Harry, for example, had Hedwig as his pet, but an owl wouldn’t necessarily be closely associated with him, because very many wizards had pet owls. Dumbledore’s having a pet phoenix was a more significant achievement. The note on the phoenix’s classification explains that few wizards have successfully domesticated one. Therefore, having a phoenix would be more representative of Dumbledore than the average wizard’s pet would be of them, both because of their close relationship and its rarity.
“11 The phoenix gains a XXXX rating not because it is aggressive, but because very few wizards have ever succeeded in domesticating it.”
- Fantastic Beasts and Where to Find Them
While there may or may not be some deeper out-of-universe symbolism intended by J.K. Rowling in Dumbledore having a pet phoenix, it seems likely that in-universe, his Patronus is a phoenix because of his close relationship with Fawkes.