N.B For Future Reference - There is a better solution below involving Tags

You need a DataType of Product - Checkbox list - which is a checkbox list with you website names within them.

And your Product Document Type must have the datatype.

Here is an example Content Tree (I have placed the product in a Container DocumentType name Product Repository

N.B Product2 does not have Website1 selected

The Website DocumentTypes contain the following code

<h1>Products Avaialable on whidh Site (@Model.Content.Name)</h1>
<a id="getProducts" href="javascript:void(0);">Get Products</a>
<div id="productList">
    <span>see below</span>
    <ul></ul>
</div>




<script type="text/javascript">
    $("#getProducts").click(function () {
        GetProducts();
    });

    function GetProducts() {
        $.ajax({
            url: '/umbraco/Surface/ProductSurface/ByWebsite?websiteName=' + encodeURIComponent('@Model.Content.Name'),
            error: function (data, textStatus, jqXHR) {
                alert(data.responseText);
            },
            success: function (data, textStatus, jqXHR) {
                for(var i=0;i < data.products.length; i++) {
                    $('#productList ul').append('<li>' + data.products[i] + '</li>');
                }
            }
        });
    }
</script>

and the SurfaceController is here

public class ProductSurfaceController : SurfaceController
{
    public ActionResult ByWebsite(string websiteName)
    {
        //Get ContentService
        var contentService = ApplicationContext.Services.ContentService;
        //Get DataTypeService
        var dataTypeService = ApplicationContext.Services.DataTypeService;
        //Get UmbracoHelper
        var umbracoHelper = new UmbracoHelper(UmbracoContext.Current);
        //Get Root Items
        var rootContent = contentService.GetRootContent();
        //predicate - reusable - Not Name = "Product Repository" - you could pass this in
        bool predicate(IContent c) => c.Name.Equals("Product Repository", StringComparison.OrdinalIgnoreCase);
        if (rootContent.Any(predicate))
        {
            //Get Product Repository and Render Template
            var repo = rootContent?.Where(predicate).FirstOrDefault();
            var products = repo.Children().Where(c =>
            {
                //Get Product Taxonomies Property
                var properties = c.Properties.Where(p => p.Alias.Equals("productTaxonomies", StringComparison.OrdinalIgnoreCase));
                if (properties.Any())
                {
                    //If Exists Get Datatype
                    var property = properties.FirstOrDefault();
                    var dataType = dataTypeService.GetDataTypeDefinitionById(property.PropertyType.DataTypeDefinitionId);
                    //Get Pre Values of Datatype
                    var preValues = dataTypeService.GetPreValuesCollectionByDataTypeId(dataType.Id).FormatAsDictionary().Values;
                    var values = property.Value.ToString().Split(',').ToList();
                    //Compare Selected Valued Within Content Item and Website Name and Return Item If True
                    return preValues.Where(pv => values.Contains(pv.Id.ToString()) && pv.Value.Equals(websiteName, StringComparison.OrdinalIgnoreCase)).Any();
                }
                return false;
            }).Select(p => umbracoHelper.RenderTemplate(p.Id, p.Template.Id).ToString()).ToList();                
            return Json(new { products }, JsonRequestBehavior.AllowGet);
        }
        return new EmptyResult();
    }
}

which gives me the following output