The menu does not have to do with the icon inside the general settings menus.

These setting menus are created from JS and is looking at the icon.png file inside "module_name/static/description/icon.png" inside the module.

_getAppIconUrl is the method of JS to create the icon URL. You will have to override this method and add the path of your icon. You can add the condition to change the icon of a specific module(s).

Try the following code in your custom module:

Ex:

odoo.define('custom_base.settings', function (require) {
"use strict";

var BaseSettingRenderer = require('base.settings').Renderer;

BaseSettingRenderer.include({

    _getAppIconUrl: function (module) {
        if (module == 'website') {
            return "/my_website_module/static/description/icon.png";
        }
        else {
            return this._super.apply(this, arguments);
        }
    }
});
});

I hope this will solve your problem.