Is expiation (kaffarah) for an oath obligatory upon one who swore not to open the door until he knew who was knocking, and then it became clear that the knocker was not the person intended by his oath?

If your intention when swearing the oath was to specify your cousin (daughter of your maternal aunt), then you do not break your oath by opening the door to someone else, and there is no expiation for breaking an oath upon you. Oaths are based on the intention of the one swearing them, then on the reason and motive for the oath. So, whenever one intends something that their oath can encompass, their oath is tied to what they intended, not to what they uttered. If there was no specific intention, then the oath is specified and restricted by the reason that prompted it, as this serves as evidence of the intention and purpose.