Problems are arising because objects do not compare as equal just because they have the same property values.

let oldItems = JSON.parse(localStorage.getItem('newProduct')) || [];

creates oldItems as an empty array or an Array of newly created objects with property values representing previous items clicked and saved.

let data = JSON.parse(localStorage.getItem('newProduct'))

will also create an array of new objects for previous buttons clicked, if there were any, but won't contain the same objects as oldItems because JSON.parse creates new objects each time it is called.

Defining newItem also creates a new object which will not be the same object as any of the entries in oldItems or data.

To solve the problem you could add unique product codes to all items, so you can search previous data for a match based on product code alone, or write a comparison routine to check all the properties that need differentiating (such as color).

Personally I would try to get a product-id added into page data, search for it among previous items, and if product-ids match, then then the item has been been clicked before (but the customer could be ordering a different color).

Be wary of using the button loop counter as an id value, or even the image source string - it's easy to imagine ways they could go wrong or be compromised by page changes outside your control.

So thanks for the help I've received from @traktor, I was able to come up with a solution to this problem. I've added a data-prodid to each product and then pass it as 'id' to localstorage. Then I check if there's any element in my array that contains the same ID i'm clicking into and if there's any result, then an alert gets executed. Otherwise, the element gets saved.

let addCartItemButtons = document.getElementsByClassName('product-description-add')
for (let i = 0; i < addCartItemButtons.length; i++){
    let button = addCartItemButtons[i]
    button.addEventListener('click', function(event){
        let buttonClicked = event.target
        let getProdId = buttonClicked.parentElement.parentElement.parentElement.getAttribute('data-prodid')
        let getTitle = buttonClicked.parentElement.parentElement.parentElement.querySelector('.product-title').innerText
        let getImage = buttonClicked.parentElement.parentElement.parentElement.querySelector('.product-header img').src
        let getColor = buttonClicked.parentElement.parentElement.querySelector('.product-description-text li span').innerText
        let getSize = buttonClicked.parentElement.parentElement.querySelector('.product-description-text li select').value
        let getPrice = buttonClicked.parentElement.parentElement.querySelector('.product-description-price').innerText
        let getSpan = buttonClicked.parentElement.parentElement.querySelector('li span').getAttribute('id')
        let oldItems = JSON.parse(localStorage.getItem('newProduct')) || [];
        let newItem = {
            'id': getProdId,
            'title': getTitle,
            'image': getImage,
            'color': getColor,
            'size': getSize,
            'price': getPrice,
            'spanid': getSpan,
        };
        
        if(localStorage.getItem('newProduct') == null) {
            oldItems.push(newItem);
            localStorage.setItem('newProduct', JSON.stringify(oldItems));
        } else {
            let data = JSON.parse(localStorage.getItem('newProduct'))
            let idCheck = data.filter(x => x.id === newItem.id).map(x => x.foo);
            let idCheckResults = idCheck.length
            if(idCheckResults > 0){
                alert('element already added')
            }else{
                oldItems.push(newItem);
                localStorage.setItem('newProduct', JSON.stringify(oldItems));
            }
        }
        let windowCartProducts = JSON.parse(window.localStorage.getItem("newProduct"));
        let productsInCart = document.getElementById('cartProducts')
        let productsAdded = 0
        for(let i = 0; i < windowCartProducts.length; i++){
            productsAdded = windowCartProducts.length
        }
        productsInCart.innerHTML = productsAdded + `<i class="fas fa-shopping-cart"></i>`
    })
}