How to handle multiple enemies with the same script?

The start of your reasoning is good, but you get lost when trying to reduce the enemy HP. Let's restart from the Sword:

In your Sword script, you have a function OnTriggerEnter2D. In this function, you can check if the collider is of type "Enemy", and, if so, apply the "damage" function:

void OnTriggerEnter2D(Collider2D collider)
{
    GameObject objectCollided = collider.gameObject;  // Get a reference to the object hit
    if (objectCollided.CompareTag("Enemy")) // If the object is an enemy
    {
        objectCollided.getComponent<EnemyController>().doDamage(swordStrength); // Here you call the enemy function that will lower his health
    }
}

A more advanced solution would be to generalize this behavior for any objects that are damageable. Let's imagine that, in the future, you want to add destructible items, like a destructible wall. This wall would have HP, get damaged, and die when its HP gets to 0, just like an Enemy.

This is achievable via a script component "Damageable" that you will add to any prefabs you have that are damageable:

public class Damageable : MonoBehavior
{
    public int maxHP;       // This has to be set in the inspector
    private int currentHP;

    void Start()
    {
        currentHP = maxHP;
    }

    void doDamage(int damage)
    {
        currentHP -= damage;
        if (currentHP <= 0)
            Destroy(gameObject);
    }
}

By attaching this script to a prefab, it becomes damageable. Then, in your Sword script:

void OnTriggerEnter2D(Collider2D collider)
{
    GameObject objectCollided = collider.gameObject;  // Get a reference to the object hit
    Damageable damageableComponent = objectCollided.getComponent<Damageable>();

    if (damageableComponent) // If the object is damageable
    {
        damageableComponent.doDamage(swordStrength); // Here you damage the object, without knowing which type it is
    }
}

I hope this helps.