The effects of the wrathful smite spell can apply to anything, including objects; the phrase in question makes it so that only creatures make the Wisdom saving throw
Most of the smite spells (all but thunderous smite and wrathful smite) begin with the following:
The next time you hit a creature with a melee weapon attack...
The wrathful smite spell, however, states:
The next time you hit with a melee weapon attack during this spell's duration, your attack deals an extra 1d6 psychic damage. Additionally, if the target is a creature, it must make a Wisdom saving throw or be frightened of you until the spell ends...
Most smite spells only have effects against creatures, however, the effects of wrathful smite are not restricted in this way so they can apply to anything you hit, including objects.
The additional 1d6 psychic damage applies no matter what you hit; however, the wording in question is there so that the Wisdom saving throw is only made if the target is a creature.
Of note, there are things which are neither creatures nor objects and they are explicitly called out as such. The only one I know of is the Shepherd Druid's Spirit Totem which states:
The spirit creates an aura in a 30-foot radius around that point. It counts as neither a creature nor an object, though it has the spectral appearance of the creature it represents.
Whether this being counts as "living" would be up to a GM (though I don't know any game-effect where it would matter). However, you can't actually hit the spirit totem, it has no stats. If you could then it would take the additional 1d6 psychic damage but it would not make the Wisdom saving throw as it is not a creature.
The fact that there are extremely few things which are neither creature nor object is supported by the rules constant use of the phrase "creatures and objects" such as in the see invisibility spell which states:
For the duration, you see invisible creatures and objects as if they were visible
This (and many other instances of this phrase throughout the rules) help show that unless stated otherwise something must be either a creature or an object (or both, potentially, see this question for that discussion: "Is there anything that is simultaneously a creature and an object?")