Is it obligatory to inform a friend that two pieces of a product bought for him online have arrived, after paying for only one piece and the seller agreeing to keep both, or is it permissible to keep the additional piece without his knowledge?

If you bought an item for your friend as an agent, and the seller gifted an extra piece, then this piece belongs to your friend (the principal); because the rights of the contract revert to the principal, not the agent. This increase is considered a reduction in price.