If your product has a name and logo but is not trademark registered or part of a recognized brand, you can list it as a generic product. However, ensure the product's name or logo does not imply brand ownership or cause confusion, as this may lead to intellectual property complaints.

Amazon's policy requires all products to be accurately listed. Using "Generic" in the brand name field is acceptable for unbranded items, but ensure compliance with the Intellectual Property Policy to avoid potential listing issues or penalties.

For customized products, it’s advised to clearly explain in the description that it’s unbranded or customized. If you plan to scale, consider trademark registration to establish your brand officially on Amazon.