Thank you for reaching out and it's great to hear that you're planning on opening a store and selling online. Shopify cannot tell you definitively what requirements there are for running a business and selling products in your jurisdiction, as these will of course vary from country to country and region to region, but in general merchants are fine to start selling without needing to register their business when they are just starting out. I would recommend doing your own research though as I am not familiar with business laws in Greece and it is best practise to know what your obligations are from a tax and registration perspective from the start.
One thing to consider is whether or not you need to register for a VAT number, as we may ask for this information in compliance with EU tax regulations. You can read more about this on third-party blogs such as this one, but if after doing your own research you are still not sure of your obligations, I would recommend consulting with a tax professional to learn more.
Victor | Shopify Social Care