Is “A Bird in the Hoof” a reference to “Harry Potter and the Chamber of Secrets?”

In Harry Potter and the Chamber of Secrets chapter 12, Harry Potter has a meeting with the most powerful wizard Headmaster Dumbledore, the one who everyone either respects or fears. In the meeting, but when Professor Dumbledore isn’t present, Harry meets his pet bird. The bird looks ill and eventually apparently dies, bursting into flames. Harry gets very scared because he’s afraid that Headmaster Dumbledore will blame Harry, even though Harry actually has no intention to hurt the bird.

Standing on a golden perch behind the door was a decrepit-looking bird which resembled a half-plucked turkey. Harry stared at it and the bird looked balefully back, making its gagging noise again. Harry thought it looked very ill. Its eyes were dull and, even as Harry watched, a couple more feathers fell out of its tail.

Harry was just thinking that all he needed was for Dumbledore’s pet bird to die while he was alone in the office with it, when the bird burst into flames.

Harry yelled in shock and backed away into the desk. He looked feverishly around in case there was a glass of water somewhere, but couldn’t see one. The bird, meanwhile, had become a fireball; it gave one loud shriek and next second there was nothing but a smouldering pile of ash on the floor.

Just then, the Headmaster arrives. Just like in a magic trick, the bird reappears alive. The Headmaster reveals to an astonished Harry that the bird is a phoenix, and what Harry saw is an expected part of its lifecycle. Harry is off the hook this time.

The office door opened. Dumbledore acme in, looking very sombre.

‘Professor,’ Harry gasped, ‘your bird – I couldn’t do anything – he just caught fire –’

To Harry’s astonishment, Dumbledore smiled.

‘About time, too,’ he said. ‘He’s been looking dreadful for days. I’ve been telling him to get a move on.’

He chuckled at the stunned look on Harry’s face.

‘Fawkes is a phoenix, Harry. Phoenixes burst into flame when it is time for them to die and are reborn from the ashes. Watch him …’

Harry looked down in time to see a tiny, wrinkled, new-born bird poke its head out of the ashes. It was quite as ugly as the old one.

Let’s skip ahead a decade. In My Little Pony: Friendship is Magic S1 E22 “A Bird in the Hoof”, Fluttershy is invited to a formal dinner with Princess Celestia, the ruler of all Ponyville with 100% approval rating. Fluttershy is introduced to Princess Celestia’s pet bird Philomena, an ugly little bird that looks very ill. Fluttershy tries to nurse the bird to improve its health, thinking that the Princess is too busy to look after it, but nothing she tries works. Despite all her good intentions, the bird eventually apparently dies and bursts to flames mid-air.

On cue, as Fluttershy stands holding the bird’s ashes in her hooves, Princess Celestia appears. Fluttershy is very afraid that Princess Celestia will blame her for the pet’s death, and tries to apologize. But then, the bird rises anew from the ashes. Princess Celestia reveals that Philomena is a phoenix, an exotic magical bird, and being reborn from ashes is normal for it. Fluttershy is off the hook this time.

“All just a normal part of the life cycle of a phoenix.”

So my question is. Is the episode intended to be a reference to Harry Potter and the Chamber of Secrets?