There is 1 unbreakable weapon in the game, but it has a caveat.
The Sword that Seals the Darkness (also known as the Master Sword) is effectively unbreakable. However, while using it, it will eventually "run out of charge" and can't be used for around 10 minutes, but it will remain in your inventory. The only exception to this is when the blade glows Blue. This means it is near enemies that the game considers to be greatly corrupted. While the blade is glowing blue, it has massively increased durability when striking these enemies. However, it will take normal durability damage when striking rocks/normal enemies/etc, even while glowing blue. It's damage also doubles from 30 to 60.
How to obtain it:
You must go to the large Woods shrouded in fog north of Hyrule Castle. Find your way through them (you cannot paraglider in) until you reach Korok Forest. Sitting in the middle of the forest will be the Master Sword. In order to pull it from the pedestal, you need 13 hearts, as pulling the sword out drains 12 and 3/4 hearts. Temporary hearts do not count for this, you need a full 13 red hearts.
There are a couple of other breakable weapons that are infinitely obtainable.
Each Divine Beast you defeat will reward you with a weapon that represents the Guardian who controls the beast. These can all be rebuilt by NPCs in the city where you obtained them when they break. They all require one generic weapon of the same type, 5 flint, and 1 diamond.
According to this article you should have gotten it on your first use of the amiibo. And if you did not put Epona into a stable you may not get her to summon again. It does not seem that there are any requirements for getting her to summon other than using the amiibo and it being the first time doing so, there are no indicators that using it at a specific point in the game changes how this works.
If the player has the Smash Bros. Series Link Amiibo, you can use the
Amiibo Rune to tap Link onto the NFC reader. While most other Amiibo
will summon a supply drop of items - the first time the Smash. Bros
Series Link is used, Epona will be summoned instead.
So my guess is, based off their explanation, is that you potentially summoned her and didn't notice and it may be stuck where you can't summon her again. Or you summoned her in the Plateau and you did not get to put her in a stable, thus she disappears.
However, if you wish to keep and ride Epona - you must register the
horse at a stable as soon as possible. If you summon Epona in a place
where you cannot get to a Stable, like on the Great Plateau, you will
be unable to summon Epona again, as using the Smash. Bros Series Link
Amiibo on other days will only summon standard supply drop items.
However; there is another article that mentions a fountain that may be of help to you.
The downside, of course, is that if Epona dies -- that's it. You
cannot respawn Epona by tapping the amiibo again after she dies, and
you will only be able to get her by restarting your game or by finding
the Horse God's Great Fairy Fountain.
So perhaps there is a chance you can get Epona back, based off of what that article says:
Unlocking this Great Fairy will allow you to bring any horse back to
life -- provided that it was registered, and that you have the means
to pay for it. Even legendary or rare horses can be brought back to
life at this fountain -- including Epona (provided you already
summoned her with an amiibo).
But since it doesn't seem like you registered her if she did spawn by chance, it may not work either. The only solution remaining after that, as far as I can tell, is restarting, which I can't imagine you want to do.
Best Answer
To unlock Master Mode, you must purchase and download The Master Trials (DLC pack 1). After doing so, you should see an option to start "Master Mode" on the main menu.
Selecting Master Mode for the first time will start you with a new game. It is not possible to switch your current save to master mode. However, starting in Master Mode will not clear your current save. You will be able to toggle between your normal mode save and your master mode save on the main menu.