As you point out in your own answer, Euron used both a personal Coat of Arms and the one belonging to his House.
I won't report references as they would only duplicate what you wrote in your answer, and I'll focus on the other doubt that you raise in your question, whether Euron was legitimately allowed to use his Family's emblem.
We should consider that he was not the only exile that we know of, we have some more examples about characters that were banished/exiled from their homeland; we can try to find some common pattern:
Jorah fled Westeros to escape execution for participating in slave trade. He was still wearing his House's Arms in his self-imposed exile:
He [Jorah] was an older man, past forty and balding, but still strong and fit. Instead of silks and cottons, he wore wool and leather. His tunic was a dark green, embroidered with the likeness of a black bear standing on two legs.
A Game of Thrones - Daenerys I
He was exiled by the Mad King Aerys II during Robert's Rebellion, and Robert himself was not willing to recall him, being one of Rhaegar's closest friends and supporters; while in exile he concealed his identity as Griff, and hid anything that could sell off his disguise; but when he came back to Westeros and abandoned his disguise, he openly displayed his Arms:
Griff had let his beard grow out during the voyage, for the first time in many years, and to his surprise it had come in mostly red, though here and there ash showed amidst the fire. Clad in a long red-and-white tunic embroidered with the twin griffins of his House, counterchanged and combatant, he looked an older, sterner version of the young lord who had been Prince Rhaegar's friend and companion … but the men and women of Griffin's Roost still looked at him with strangers' eyes.
A Dance with Dragons - The Griffin Reborn
He fled from Westeros after his Trial accused him guilty. He had a bounty on his head so he concealed his true identity, using the names Hugor Hill and Yollo; we can safely assume that he was not openly displaying his family Coat of Arms, bu he did it for the sake of safety, there is no evidence that he was stripped (or willingly renounced) the right to wear them.
There are other characters that were not strictly exiled, but in a way or another were considered outlaws, so they loosely shared the status of banished individuals: in example Beric Dondarrion, Sandor Clegane and Brynden Tully; despite being outlaws, they were openly displaying their House or personal Arms, or did nothing to conceal them (which could suggest that it could be forbidden to wear them). It seems that even in this case, when they were not displaying them was more for not wanting to be recognized rather than a legal prohibition.
It seems that the status of exile or banished individual does not imply a prohibition to use one's House Coat of Arms. We have example of oucast characters from through all Westeros that carry them without much concern.
While Iron Islands have a very distinctive culture, very different from the rest of Westeros in many respects, we don't have any indication that concerning Heraldry they have different rules. We can assume that the general trend could be valid even there.
As you point out in your answer, he is known to carry a personal Coat of Arms, but this does not mean that he was forbidden to carry his House's: we know of a long list of characters who had personal arms for many different reasons: being cadet sons, being bastards, pure vanity, and so on.
Moreover, Euron does not seem to be the person who could suffer to be imposed any kind of restriction; even if he was forbidden to use the Emblems of House Greyjoy, it could have be perfectly in-character from him to use them just for the sake of taunting and defiance.
It should also be considered that since after the Kingsmoot, there was nobody who could de facto challenge him; even if there was in fact a rule about this, nobody could impose him to renounce to them.
Having a personal Coat of Arms, then, does not prevent him to use it in conjunction to his House's, he could have used both of them depending on the situation: his personal one for personal matters and
to disguise himself as Sauron to be identified as Euron himself, the Greyjoy's in his role of Leader of that House and as the Iron King.