First, let's talk about what's possible with two of the most commonly used WooCommerce organizational tools: categories and tags. Each product in WooCommerce can have categories and tags. Categories tend to be hierarchical while tags have a flat structure. It's a best practice to have one category and several tags.
Here's an example product's category and you could add tags right below:
You might notice that you can add multiple categories to a product but that doesn't mean you should. I recommend you only use one category for each product. WooCommerce with the Storefront theme and a few other themes has a really nice breadcrumb feature that shows users where they are in the catalog. This won't work well with multiple categories.
You can see the breadcrumbs in the following screenshot:
The breadcrumb feature on the...