What equipment is unique?
The following items can only be obtained in one way:
- Master Sword
- Hylian Shield
- All the Champion weapons
- Cerimonial Spear
- Rubber Armour Set
- Barbarian Armour Set
- Climbing Armour Set
- Zora Armour Set
- Old Shirt
- Old Trousers
- Snow Boots
- Sand Boots
- Thunder Helm
"Of the Wild Armour Set"
All items that can only be obtained in one place (except the Nintendo Switch Shirt) and don't respawn can be re-obtained either by remaking them (in the case of the Champion weapons), or can be purchased from a special store.
Spoilers for the store:
You must complete the quest "From the Ground Up", which is unlocked after purchasing the house in Hateno Village. After you've fully built the town, and NPC will spawn on the roof of the building right of the entrance of Tarry Town. Speak to him. He can sell anything that can only be normally obtained once in the game, bar the Master Sword, Cerimonial Spear, and Champion Weapons
To the best of my knowledge, the only item that cannot be re-obtained is the Master Sword. To accommodate this, it is not droppable and has a different way of handling durability.
All other items are re-obtainable (although you can't necessarily have multiple at a time). To your examples:
- Champion weapons can be re-gotten with a diamond and some other materials
- The Hylian Shield can be re-bought somewhere in the game if it breaks after being found initially
- the Frost Spear can be found in the Coliseum Ruins, supposedly, so this is likely not a unique item.
The Ceremonial Trident also can only have one instance in the world at a time, as well.