The Alter Rebbe is referring to a case where a person fulfilled his obligation to make Kiddush, and before he makes an after-blessing, another person arrives. In that scenario the person who already made Kiddush should not say it again, because the law of ‘Arvut’, (mutual obligation,) does not come into play if the second person is able to make his own Kiddush.
When a person comes to his home and wishes to drink wine he needs to make the blessing, and thus he can make Kiddush for his wife. This is what I remember as being the practice of Rabbi Dvorkin in 770 and other Rabbis in the area. Rabbi S. Z. Auerbach also maintains that this is the correct practice, (Halichos Beisoh 15:139). “Whoever has fulfilled their obligation of Kiddush by day can make Kiddush for another, even though for him it is only a matter of making the blessing over wine, since that is the essence of Kiddush – the blessing over the wine, and it is a similar obligation.” He concludes that that is the accepted practice.