Cute idea!

First of all, some ideas to simplify and optimise your code:

1. Don’t use a separate variable for catCount when you already have the List‘s cats.Count value. It’s completely redundant and adds a bunch of extra code to sync these values.

2. Don’t keep deleting and reinstantiating the cats from the cats list. Rather, when a cat is picked up, disable the object and add it to the list. Then when it is put down, activate it and remove it from the list. You already have the GameObjects instantiated and referenced there. If you really need a reference to all cats at any given time, then do this with a new list (or an array if the total amount doesn’t change) called allCats instead.

3. I can’t imagine why, in your OnTriggerStay method, you’d be creating a new instance of the object you collided with, then disable it, then destroy the original. That makes little to no sense to me. Perhaps don’t do this.

It seems to me that you need to do some research on what it means to instantiate an object. Generally objects are instantiated at load time, then simply activated and deactivated as needed during run time. This method is way more performant and helps alleviate the issue of accidentally creating memory leaks (filling up your RAM with tons of object instances).

This is not as important, especially since you are learning and prototyping, but try to extract things into methods where you can. You could put the code for picking up and dropping cats into PickupCat(GameObject cat) and DropCat() methods. This really improves the readability and reusability of your code.

So, refactoring your code based on the sugestions (plus a few extra touches):

using System.Collections.Generic;
using UnityEngine;
using UnityEngine.UI;

public class PlayerCatching : MonoBehaviour
{
    public Text playerScore;
    private int playerScoreInt;

    public Text catHoldCount;
    public List<GameObject> cats;
    private const int maxHeldCats = 3;

    private void Start()
    {
        playerScoreInt = 0;
        cats = new List<GameObject>();
        playerScore.text = playerScoreInt.ToString();
    }

    private void Update()
    {
        playerScore.text = playerScoreInt.ToString();
        catHoldCount.text = cats.Count.ToString();

        if (Input.GetKeyDown(KeyCode.F))
        {
            DropCat();
        }
    }

    private void OnTriggerStay(Collider other)
    {
        if (Input.GetKeyDown(KeyCode.E) && other.gameObject.CompareTag("Cat"))
        {
            PickupCat(other.gameObject);
        }
    }

    private void PickupCat(GameObject cat)
    {
        if (cats.Count >= maxHeldCats)
            return;

        playerScoreInt++;
        cat.SetActive(false);
        cats.Add(cat);
    }

    private void DropCat()
    {
        if (cats.Count <= 0)
            return;

        GameObject releaseCat = cats[cats.Count - 1];
        Vector3 catReleasePosition = new Vector3(transform.position.x, transform.position.y + 1, transform.position.z);
        releaseCat.transform.position = catReleasePosition;
        releaseCat.SetActive(true);
        cats.remove(releaseCat);
    }
}

Note that I changed the input for E to GetKeyDown instead of GetKey. I also added a constant for the maxHeldCats to replace your hardcoded value. You also had unnecessary using directives.

That should not only simplify reading the code, but also the execution, because now your PC doesn’t have to create and destroy objects the whole time, it just manipulates what is already there.

As a final note, I would recommend the variable names be changes to something clearer:

Text playerScoreText;
int playerScore;

Text catHoldCountText;
List<GameObject> heldCats;

I hope this helps. Let me know if and how this works for you please!