I'm a little bummed that Blizzard doesn't offer Haedrig's gifts for both Softcore and Hardcore. I understand that they are giving a full set to jump start the grind, but I specifically have saved my gift thinking I would use it on a Hardcore character.
as far as I can discover, the "mail" system is separated between Softcore and Hardcore so that when you reach an accomplishment (chapter2/3/4) you can only receive it on the "core" you are playing.
This doesn't make sense to me - it wasn't documented when I read the season journey notes (unless I completely missed it).
The reason I really don't like the above answer - I had no problem receiving the challenge rift cache on Softcore side one week and on Hardcore side another.
oh well - maybe that will change in the future, and I do like the helltooth set with an angry chicken.
Every single aspect of hardcore is distinct from softcore. It's always been like that. No shared items, gold, blood shards, paragons, or even transmogs. The only thing I'm not sure is the modes. I can't remember if adventure is unlocked from the start if you've unlocked it in softcore. Same with the campaign journey.
So, sorry to break it to you, but not sharing the gift between SC and HC does, in fact, make sense .
With that said, that also means you can get the gift on your HC character, mind you. You just need to do the journey again with them.