There are random items landing in my inventory. It happened in the very beginning of the game, but back then I did not know my ways around yet, so I did not care. Back then it was one amnesia potion, now, very recently, it is one mysterious bag. In both cases there is no message, saying I have been given a gift and there is nothing in the game log. Also I am not aware of any achievement or any privilege, allowing me to be gifted at random. I am not sure, where this comes from or what causes it and I would like to have some clarification. Other players are also encouraged to share their experiences, should they present any such like.

Thanks to Seelenblut, the origin of the bag have been successfully explained.

The mysterious bag is a christmas gift, just look at the new update news  ;D

You get the things like amnesia potions or some strange fruits from the exploring, they are sometimes in chests just take a look at the exploration log, I'm sure they don't appear in your inventory :D

I know the chests on islands can contain such items, but it confused me, because I have not found any notice neither in the exploration log, nor anywhere else. 

Your pets can obtain different items from quests. These logs are kept for 2 days and available in your Reports.