Cannot find "self" in scope & Consecutive declarations on a line must be separated by ";".

Isn't SELF supposed to be pointing to GestureManager class?

Yes it is… But …

Code Block 
 let rightSwipe = UISwipeGestureRecognizer(target: self.vc, action: #selector(respondToSwipeGesture(_:)))
    rightSwipe.direction = .right
    let leftSwipe = UISwipeGestureRecognizer(target: self.vc, action: #selector(respondToSwipeGesture(_:)))
    leftSwipe.direction = .left

This code should be inside a func (like viewDidLoad), not free floating in the class, where it is looked at as a declaration.

So you should have:

Code Block 
     override func viewDidLoad() {
          
          super.viewDidLoad()
          let rightSwipe = UISwipeGestureRecognizer(target: self.vc, action: #selector(respondToSwipeGesture(_:)))
          rightSwipe.direction = .right
          let leftSwipe = UISwipeGestureRecognizer(target: self.vc, action: #selector(respondToSwipeGesture(_:)))
          leftSwipe.direction = .left
}

Why is this happening?

Because you are using self in the initial values of instance property declaration.
And you are writing executable statements where they are not allowed.

why self cannot be found in scope? 

You cannot use self in the initial values of instance property declaration in Swift.

Isn't SELF supposed to be pointing to GestureManager class?

Swift utilizes case-sensitive identifiers. You should better care about letter cases. If you want some words to be emphasized, you can use some markdown features.

---

But the most critically bad thins is this one:

 I'm expecting to pass the ViewController to be used in the "target" attribute

When you use target-action pattern, the target and the action needs to be consistent.
If you pass self.vc to target:, you need to pass an instance method of self.vc to action:.

---

The right fix may depend on the usage of the class GestureManager, but one possible fix would be something like this:

Code Block 
class GestureManager {
    var vc: UIViewController?
    
    @objc func respondToSwipeGesture(_ gesture : UIGestureRecognizer) {
        if let swipeGesture = gesture as? UISwipeGestureRecognizer {
            switch swipeGesture.direction {
            case .right:
                print("swiped right!")
            case .left:
                print("Swiped left")
            case .up:
                print("Swiped up")
            case .down:
                print("Swiped down")
            default:
                return
            }
        }
    }
    
    lazy var rightSwipe: UISwipeGestureRecognizer = {
        let rightSwipeGR = UISwipeGestureRecognizer(target: self, action: #selector(respondToSwipeGesture(_:)))
        rightSwipeGR.direction = .right
        return rightSwipeGR
    }()
    
    lazy var leftSwipe: UISwipeGestureRecognizer = {
        let leftSwipeGR = UISwipeGestureRecognizer(target: self, action: #selector(respondToSwipeGesture(_:)))
        leftSwipeGR.direction = .left
        return leftSwipeGR
    }()
}

• You can use self in the initial values of lazy instance property.

• To run some executable statement like rightSwipe.direction = .right, you need to create a function and call it.

• If you do want to call the method respondToSwipeGesture(_:) of GestureManager, the target needs to be self, not self.vc.

If you are not accustomed to utilizing closure, you can define explicit functions:

Code Block 
    lazy var rightSwipe: UISwipeGestureRecognizer = createRightSwipeGR()
    
    private func createRightSwipeGR() -> UISwipeGestureRecognizer {
        let rightSwipeGR = UISwipeGestureRecognizer(target: self, action: #selector(respondToSwipeGesture(_:)))
        rightSwipeGR.direction = .right
        return rightSwipeGR
    }
    
    lazy var leftSwipe: UISwipeGestureRecognizer = createLeftSwipeGR()
    
    private func createLeftSwipeGR() -> UISwipeGestureRecognizer {
        let leftSwipeGR = UISwipeGestureRecognizer(target: self, action: #selector(respondToSwipeGesture(_:)))
        leftSwipeGR.direction = .left
        return leftSwipeGR
    }