The state, condition, or characteristic qualities of being a dragon; dragonness or the quality of resembling or acting like a dragon.
From 'dragon' + '-hood' (a suffix meaning state or condition, from Old English 'had'). This combines the mythical creature with a productive English suffix used to denote abstract qualities.
The suffix '-hood' is incredibly productive in English—you can attach it to almost anything to mean 'the quality of being that thing,' which is why we have everything from childhood to falsehood to your invented 'dragonhood.'
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