Most RPGs concentrate on trade and auction.
A gift is often a trade for 0 currency points.
Instead a more complex gift system can be constructed.
An item/pet/object marked as gift,can be
1.sent anonymously: if its rejected it returns to owner(and target can block further gift without knowing the sender)
2."wrapped" in box item with other gifts.
3.cursed items can be sent as gifts too: it adds a new tactical angle to gift system.
4.left to specific location and can be picked up only by target or specific restricted group(e.g. <10 lvl).
5.Can contain custom messages or media embedded into gift box.
6.marked not for sale:target can't sell the item.
7.marked not transferable: only target can use item(or return to sender)
8.any restriction or mechanic that is cumbersome to implement/use in trade system, can be used in gift systems.

NPCs may also participate in gifting if player needs something(e.g. player lacks equipment/resources).