I noticed in the tundra the hut that gives you bandages activates the bandage button when you explore the building, even if your pet is not hurt. I've also noticed that if you click on the bandage button after this but haven't taken damage, it will use the item but it won't heal the pet because it wasn't injured.

I noticed this too, but when you click on the bandage it doesn't use it, a message just pops up saying that the pet doesn't have any wounds to heal.

Huh. That's odd. I thought it went down as well because I once used it and the number was the same as when I entered the island. Maybe its just that the number of bandages never went up?

If you use the hut bandage when your pet isn't injured, the count of your bandages should go up by one (that's how it's been working for me anyway).  If your pet is injured and you use the hut bandage, the number should not change.

This is actually works as designed. Every camp has a stock of medical kits and other necessary supplies that can be used by a traveller (though everything except medkits is spoiled long time ago or has no use for pets), or picked up in order to be used later in a case of emergency. It doesn't know in advance if a certain traveller will need emergency healing or not. In game terms, the camp gives you 1 bandage and provides a chance to use it immediately. If your pet isn't injured, the bandage won't be used and will get to your inventory, and you will get a message that the pet isn't injured.