Are 3DS download codes region-locked?

Yes, they're Region Locked.

Fun fact: Because of the Region Locking, every game is considered as a different game in every Region by Nintendo.
For example: There is a Mario game released in America, Japan, Europe, and Australia, they are actually 3 different games.

Not different as in, different stuff, it's different as in, being Registered separately, got separate Approval, etc.
 
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