Your dialogue stats are also shared, charisma and stuff. Only the ones on your costumes isn't, but stuff from pots/quests is.
For the original question, don't open anything that looks like you'd want to open on your main, just drop in roster storage and switch it later. If you can't put it in roster storage, well might as well open. In general, don't open anything you don't need right now. I just leave all the boxes in my storage until I know what I want to do with them. Takes less inventory slots that way too.
A lot of stuff is shared though, the notable things not shared would be your upgrade crystals things(harmony shards) even though they look like a currency(which otherwise are shared), costumes to an extent because opening them will make them work on specific class/subclass(so like Fighter, or only for Winddancer specifically, so unless your other char is also a winddancer, can't use) and can't think of much else but probably something. Obviously bound stuff is still bound unless it's bound to roster, so generic upgrade materials from quests/una quests will be bound to that char, gear you find that is bound to you and so on, but most of the progression stuff is account wide so you build up your overall account power over time not just your main's.