UI - Does Mend allow you to create Product/Project tags and if so, how are they used?

Yes, Mend allows you to add Product/Project level tags via the settings.

Tags are used when you want to form a project naming convention corresponding to your git repositories. All it actually does (just like project/product version) is concatenating the version/tag number to the project name.
Note: It is mainly used when integrating with Kubernetes by creating project tags when scanning clusters because you can't add the "project/product" tags like you would do in CI/CD.