Unset a single item form a php array embedded in a html site with JS button

:)

The solution is rather easy, but requires some explanation in order to be understood.

What you need here is to:

1. Create a new php file, which would fetch the post data (in this case ID of an element) and then simply unset the key (sub-array), which contains the cart item you want to remove. You can use $key_to_remove = array_search($_POST['stock_item_id'], array_column($_SESSION["cart"], 'StockItemID')); and then simply unset it unset($_SESSION["cart"][$key_to_remove]);
2. Assign id="remove_<?php echo $data["StockItemID"]; ?>" to the <div class="basket-product"> and data-product-id="<?php echo $data["StockItemID"]; ?>" to the button, so you can identify it for item removal via javascript/jquery and you need that value extracted later for the item you want to remove from the corresponding session array (which is, in this case, $_SESSION["cart"]).
3. Create a callback function for removal on('click', function(){});
4. Inside that function extract that value data-product-id from the button you just clicked var stock_item_id=$(this).attr('data-product-id');
5. Inside the same function, after step 4, create an ajax call to the file from step 1 with the post data from step 4
6. On successful execution of an ajax call, delete the corresponding product row you have marked with id="remove_<?php echo $data["StockItemID"]; ?>" in step 2 with the following code $("#remove_"+stock_item_id).remove();

In the end, your code would look like this

YOUR INITIAL PHP AND HTML (With small corrections)

<?php 
 foreach($_SESSION["cart"] as $item) {
     $data = getProducts($pdo, $item);
     if ($data["ColorName"] == NULL) {
        $color = "";
     } else {
        $color = "Color: ".$data["ColorName"]."<br>";
     }
    if ($data["Size"] == "") {
        $size = "";
    } else {
        $size = "Size: ".$data["Size"]."<br>";
    }
 ?>
 <div class="basket-product">
    <div class="item">
        <div class="product-image">
            <img src="http://placehold.it/120x166" alt="Placholder Image 2" class="product-frame">
        </div>
        <div class="product-details">
            <h1>
                <strong>
                    <span class="item-quantity">
                        1
                    </span>
                    x <?php echo $data["StockItemName"]; ?>
                </strong>
            </h1>
            <p>
                <strong>
                    <?php echo $color." ".$size; ?>
                </strong>
            </p>
            <p>
                Product Code - <?php echo $data["StockItemID"]; ?>
            </p>
        </div>
    </div>
    <div class="price">
        <?php echo $data["RecommendedRetailPrice"]; ?>
    </div>
    <div class="quantity">
        <input type="number" value="1" min="1" class="quantity-field">
    </div>
    <div class="subtotal">
        <?php echo $data["RecommendedRetailPrice"]; ?> * 1
    </div>
    <div class="remove">
        <button data-product-id="<?php echo $data["StockItemID"]; ?>">
            Remove
        </button>
    </div>
 </div>
<?php
}
?>

JS

$(document).ready(function(){
    $('.remove button').on('click', function() {
        var stock_item_id=$(this).attr('data-product-id');

        $.ajax({
            url: "new_php_file_created_to_remove_item_from_session_via_ajax.php",
            data: 
                {stock_item_id : stock_item_id}
        }).done(function() {
            $("#remove_"+stock_item_id).remove();
        });
    });
});

NEW PHP FILE (new_php_file_created_to_remove_item_from_session_via_ajax.php)

<?php
    $stock_item_id = $_POST['stock_item_id'];
    $key_to_remove = array_search($_POST['stock_item_id'], array_column($_SESSION["cart"], 'StockItemID'));
    unset($_SESSION["cart"][$key_to_remove]);

    if(isset($_SESSION["cart"][$key_to_remove])) {
        return false;
    }
    return true;

For the sake of readability and further maintenance and possible additions, I would strongly recommend you to separate php, html and js code into separate files, but that's only a suggestion. :)