While under the effect of the astral projection spell, you have an astral body which has the following rules:
Your astral body resembles your mortal form in almost every way, replicating your game statistics and possessions.
I don't think that you keep stuff if you pick it normally because of the following line (but please correct me if I'm wrong):
Any damage or other effects that apply to it have no effect on your physical body, nor do they persist when you return to it.
It doesn't specifically say that you cannot pick up loot and take it with you, but it differentiates the astral form from the physical form in a way that they are in no way connected (except the cord).
But, my possessions are being replicated, including my bag of holding. If I pick something up in the Astral Plane and put it into my bag of holding, will it remain in there when I return to my physical body?