It has multiple uses.
The Sword of Wounding, (Basic Rules), the Nycaloth (MM), the Bearded Devil (Basic Rules), and the Horned Devil (Basic Rules) all inflict wounds that cause damage in subsequent rounds, however, a Wisdom (Medicine) check can stop the wound.
An example from the sword (the others are substantially similar):
…the wounded creature, or a creature within 5 feet of it, can use an action to make a DC 15 Wisdom (Medicine) check, ending the effect of such wounds…
As has been noted, in the Basic Rules, it says you can use a Wisdom (Medicine) check to staunch wounds.
The docent, from WGtE, can use a Wisdom (Medicine) check on its Warforged host, and gets significant plusses doing so.
The optional rules for lingering injuries in the DMG cite a Wisdom (Medicine) check as a way to heal a festering wound.
But, wait, it isn’t just for first aid!
In Curse of Strahd, Rise of Tiamat, Tales of the Yawning Portal, Tomb of Annihilation, and Waterdeep Dungeon of the Mad Mage, it is used to identify cause of death.
In Rrakkma, it is used to identify brown flecks as blood.
In Princes of the Apocalypse, it is used to identify bone flutes as humanoid.
It can be used to remove a control gem from a slaad.
So, although the Basic Rules just cites using a Wisdom (Medicine) check for stabilizing a dying companion or diagnose an illness, examples show that when there’s anything involving anatomy, first aid, surgery, forensics, nursing, or pathology, then a Wisdom (Medicine) check might be just what the doctor ordered.