Is it permissible for a partner to take insignificant items from the shop without the permission of his partner, knowing that the other partner has given him permission to do so, and is this considered a breach of trust?

Your friend's statement is incorrect. As long as the partnership in the shop is limited to just the two of you, and you have given your permission for this, then the right here belongs to the owners of the capital—who are only the two of you. Whatever you both willingly agree to take, there is no transgression against the right of Allah Almighty in taking it, nor is there a betrayal of trust. Such transgression and betrayal only occur when something is taken without the knowledge of the partner.