Hey there, @NewBee2
Bo here from Shopify Support!
As this is your first post in the Shopify Community I would like to welcome you! Welcome! I would recommend reading our guide on getting support from the Shopify Community here so you are best equipped when looking for support through these forum boards. We also have this great introduction to Shopify guide which I would highly recommend. What is it that your business does?
Registering with Shopify does not give you all the rights to a name, you will still need to register your business with your local authorities and go about copyrighting your name, logo, ideas, etc. We have a great guide on Copyright and Infringement here that I would recommend reading. his information doesn’t constitute professional legal advice. Consult with independent legal counsel for information specific to your region or country and circumstances.
All the Best,