The thing about putting TMC first because of the hat...That doesn't explain the tunic. I find it much more likely that OoT came first, and then people on Outset (and I assume other islands with humans, like Windfall. Considering we never see anyone at Link's age, we can't tell, though) began wearing his outfit to honor him. The tradition may have been kept in New Hyrule, but the hat was lost over time. When TMC Link got his hat at the end of TMC, the following Links may have worn their hats to honor him, and the tunics for the HoT.

Also, it's entirely possible for OoT Child Link to be a hero in the Child Timeline, and thus the Hero's Shade. The bow is called the Hero's Bow, for example, and people are always referring to an ancient hero (although that could be anyone, heroes always seem to be Links). For example, Link in OoT warned the royal family of what Ganondorf was going to do, thus bringing us to his execution in TP. Link may have been regarded as a hero for that, considering he saved everyone from living the Seven Years of Darkness. It's also possible he told his story to Zelda, who then passed it down her bloodline, until it became legend. And lastly, note the armor the Hero's Shade wears. It resembles a knight. Knights are often heroes in fantasy.

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