What plugin is needed to make this menu.
This is a variable product of WooCommerce. You don't need any special plugin for that.
Please note: If you like to assign a different fancy product for each variation please use the multistep product configurator plugin for that. (But then you won't get a dropdown but rather options to choose from like this: https://fancyproductdesigner.com/product/birthday-card-mspc/)