Can a dragon have more than one rider/loyalty at the same time? I know no rider can have 2 dragons, but say, if a Targaryen King/Queen wanted to teach their child how to ride their own dragon, could they interchange?
It doesn’t appear to be possible for a dragon to have more than one dragonrider at the same time. As far as we are aware, when dragons have had more than one rider, the new rider only came along after the previous one had died. (Balerion > Aegon, Maegor, Viserys; Vhagar > Visenya, Laena, Aemond; etc.) Note that Maegor was not a dragonrider until he was 25, since he wanted only Balerion and refused all other dragons as unworthy, and so had to wait for his father Aegon to die before he could claim him.
As for a parent and child sharing a dragon, the bond between dragon and rider is strong, to the point that a dragon may share their master’s loves and hates. (For example, the dragons of King Jaehaerys and Queen Alysanne often mated, and coiled around each other while sleeping, even long after their first masters had died.) And yet, we have the example of Syrax, Queen Rhaenyra’s dragon, when Rhaenyra’s son Joffrey attempted to ride her:
We shall not pretend to any understanding of the bond between dragon and dragonrider; wiser heads have pondered that mystery for centuries. We do know, however, that dragons are not horses, to be ridden by any man who throws a saddle on their back. Syrax was the queen’s dragon. She had never known another rider. Though Prince Joffrey was known to her by sight and scent, a familiar presence whose fumbling at her chains excited no alarm, the great yellow she-dragon wanted no part of him astride her. In his haste to be away before before he could be stopped, the prince had vaulted onto Syrax without benefit of saddle or whip. His intent, we must presume, was either to fly Syrax into battle or, more likely, to cross the city to the Dragonpit and his own Tyraxes. Mayhaps he meant to loose the other pit dragons as well.
Joffrey never reached the Hill of Rhaenys. Once in the air, Syrax twisted beneath him, fighting to be free of this unfamiliar rider. And from below, stones and spears and arrows flew at him from the hands of the rioters below, maddening the dragon even further. Two hundred feet above Flea Bottom, Prince Joffrey slid from the dragon’s back and plunged to the earth.
–The Princess and the Queen
So, that doesn’t work out, even when the other rider is the beloved child of the dragon’s master, even when they’re very well known to the dragon. A “hey mom, borrowing the dragon, be back later!” is just not possible.
However, we do know that a dragonrider may take passengers on their dragon. Visenya took young King Ronnel Arryn on a flight around the Eyrie on Vhagar, and on that same dragon Prince Aemond flew with his lover:
On the fourteenth day of the prince’s vigil, a shadow swept over the castle, blacker than any passing cloud. All the birds in the godswood took to the air in fright, and a hot wind whipped the fallen leaves across the yard. Vhagar had come at last, and on her back rode the one-eyed prince Aemond Targaryen, clad in night-black armor chased with gold.
He had not come alone. Alys Rivers flew with him, her long hair streaming black behind her, her belly swollen with child. Prince Aemond circled twice about the towers of Harrenhal, then brought Vhagar down in the outer ward, with Caraxes a hundred yards away. The dragons glared balefully at each other, and Caraxes spread his wings and hissed, flames dancing across his teeth.
The prince helped his woman down from Vhagar’s back, then turned to face his uncle.
–The Princess and the Queen
So if a Targaryen king or queen wanted to personally teach their child how to ride a dragon, they could go flying together, riding double with the child as their passenger. But if that child wanted to become a solo dragonrider, they’d have to get their own dragon to bond with – if they wanted to ride their parent’s dragon alone, they’d have to wait for that parent to die.
Hope that helps!