By default, Out of the Sandbox themes include the ability to add "new," "coming soon" and "pre-order" banners to products.
The Flex theme is our only Shopify theme that contains additional banners/badges known as "product stickers." To learn more about the additional options included in the Flex theme, check out this article.
With some custom code, it's possible to create custom banners to label products. Since this involves additional development, we would suggest reaching out to one of our recommended Shopify Experts. We have a list of some excellent developers here!