In an encounter (Wyvern Tor) my character is wielding a two handed greataxe which deals 1d12 damage. My Orc opponent was wielding the same weapon but its damage output was 2d12+6, making it much more effective. Why is there such a difference between the character's and orc's damage with the same weapon?
I find nothing in Lost Mines of Phandelver or any of the orc statblocks that explains an extra d12 or using +6 instead of +3 (many orcs are Strength 16). However, some of the specialist orcs (War Chief and Eye of Gruumsh) have a special ability:
Gruumsh’s Fury. The orc deals an extra 4 (1d8) damage when it hits with a weapon attack (included in the attack).
Volo's Guide to Monsters also has some orcs with:
Foe Smiter of Ilneval. The orc deals an extra die of damage when it hits with a longsword attack (included in the attack).
It's plausible your DM is doing a bit of homebrew, blending the two, while mistakenly doubling the Strength bonus.
In any case, the point is that monsters do not follow the same rules as player characters. They have different, sometimes special or unique, abilities that players cannot duplicate identically. In part, this is because player character abilities are designed to be repeated over the life of a character, while creature abilities are designed for things that are meant to show up once and die.
Damage output would only be equal if you were playing a game where you only encounter other creatures that are exactly as powerful as your character.
Different player characters will have different damage output, so different enemies will have different damage output.
Separately, you don't have to fight everything you meet. You may very well encounter a high-level villain very early in the game; that doesn't mean that you have to fight them the first time you meet them.
Because player characters and monsters don't use the same rules
How much damage a monster does is determined by its Offensive Rating not by what gear it is carrying.
When you design a monster (DMG chapter 9) you pick an Offensive Rating based on how much damage it puts out a round and a Defensive Rating based on how many hit points and armour it has.
For example, you pick Offensive Rating 2 which does 15-20 damage a round. You might decide to do this with 2d8+4 or maybe 1d12+6. You then pick Defensive Rating 4 which gives it AC 14 and 116-120 hit points. These two ratings average to a Challenge Rating of 3.
Unique Variant or DM Discretion for Tougher Challenge
The orc stat block has damage output of 1d12+3, which matches the strength score of 16. Running into an orc that is significantly stronger or tougher in some aspect is reflective of a higher challenge or a special variant. The damage output increase reflects the tougher challenge.