Do you only need to bow once when meeting a hippogriff?

Question

Like, you meet a Hippogriff, you bow, you come and pet him and everything's fine. But if you see the same Hippogriff again later, do you have to bow again or is it only necessary to do it once and the Hippogriff will remember that you are the same person?

Answer 1

You need to bow again.

Harry bowed to Buckbeak in the lessons:

Harry didn't much feel like exposing the back of his neck to Buckbeak, but he did as he was told. He gave a short bow and then looked up.
_{Harry Potter and the Prisoner of Azkaban, chapter 6}

but when he goes back in time he has to bow again. Remember, this is all chronologically after that lesson.

Careful not to blink, Harry stared up into Buckbeak's fierce orange eyes once more and bowed. Buckbeak sank to his scaly knees and then stood up again.
_{Harry Potter and the Prisoner of Azkaban, chapter 21}

And when Harry & Co meet Sirius and Buckbeak in The Goblet of Fire, they have to bow again:

Then, at last, Sirius slipped out of sight, and when they reached the place where he had vanished, they saw a narrow fissure in the rock. They squeezed into it and found themselves in a cool, dimly lit cave. Tethered at the end of it, one end of his rope around a large rock, was Buckbeak the hippogriff. Half gray horse, half giant eagle, Buckbeak's fierce orange eyes flashed at the sight of them. All of them bowed low to him, and after regarding them imperiously for a moment, Buckbeak bent his scaly front knees and allowed Hermione to rush forward and stroke his feathery neck.
_{Harry Potter and the Goblet of Fire, chapter 27}

And another one in the sixth book:

The great gray hippogriff, Buckbeak, was tethered in front of Hagrid’s cabin. He clicked his razor-sharp beak at their approach and turned his huge head toward them.
“Oh dear,” said Hermione nervously. “He’s still a bit scary, isn’t he?”
“Come off it, you’ve ridden him, haven’t you?” said Ron. Harry stepped forward and bowed low to the hippogriff without breaking eye contact or blinking. After a few seconds, Buckbeak sank into a bow too.
_{Harry Potter and the Half-Blood Prince, chapter 11}

Answer 2

It’d certainly be safer, but even a first bow doesn’t seem required.

In the entry on Hippogriffs in Fantastic Beasts and Where to Find Them, bowing to the Hippogriff is stated as showing good intentions, but not strictly a requirement to interacting with them. If they do bow, they can tell in advance if it’s safe to approach. Not doing it is risky, since a Hippogriff could attack like Buckbeak did with Draco, but even the first bow isn’t required. Similarly, bowing other times after first meeting and bowing to the Hippogriff wouldn’t be strictly required, but would be safer, especially if unsure if the Hippogriff is still open to their interactions.

The Hippogriff is native to Europe, though now found worldwide. It has the head of a giant eagle and the body of a horse. It can be tamed, though this should be attempted only by experts. Eye contact should be maintained when approaching a Hippogriff. Bowing shows good intentions. If the Hippogriff returns the greeting, it is safe to draw closer.
_{^{- Fantastic Beasts and Where to Find Them}}

Hermione actually climbed onto Buckbeak without having bowed to him at any point (during Hagrid’s class she’d bowed to a chestnut Hippogriff), and he didn’t hurt her or seem offended.

“Hermione put her hands on Buckbeak’s back and Harry gave her a leg up. Then he placed his foot on one of the lower branches of the bush and climbed up in front of her. He pulled Buckbeak’s rope back over his neck and tied it to the other side of his collar like reins.”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 21 (Hermione’s Secret)}}

Bowing subsequent times, then, would be a good safety measure, but not strictly required. In addition, after being around the Hippogriff for a long enough time, there’d likely be no need to keep bowing. Hagrid doesn’t bow to them when he brings his Hippogriff herd to meet his class.

“Gee up, there!’ he roared, shaking the chains and urging the creatures towards the fence where the class stood. Everyone drew back slightly as Hagrid reached them and tethered the creatures to the fence.

‘Hippogriffs!’ Hagrid roared happily, waving a hand at them. ‘Beau’iful, aren’ they?”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 6 (Talons and Tea Leaves)}}

That’s likely because they’re so used to him, that he’d no longer need to bow out of caution.