You can customize your partner's outfit by clicking on the “Gift” button next to their name in the Contacts list.

You can preview clothing that you haven’t purchased by clicking on the item. In order to buy the item for your partner, you must click once more on the item in order to trigger the purchase confirmation pop-up.

Unlike the personal avatar customization screen, there is no warning pop-up when you try to exit without having bought the items which your spouse is wearing! Therefore, make sure that you have purchased the item in order for your spouse to wear it—otherwise your spouse will default to whatever he/she was wearing before you entered the customization screen.

If your spouse is an NPC [Non-Character Player] the outfit will change right away.  If your spouse is a real-world friend, it's up to that user to dress their avatar in the clothes you have bought them. 

Note that Facial or Hair customization options can be gifted to NPCs only - not to real-life friends.