What kind of business should I register to run an online dropshipping store on shopify, and run a verified Paypal Business account?
Also I'm from Portugal (Europe).
Do you mean a legal registry? In your country you need to register based on your local laws. You're governed by local jurisdiction with internet businesses. You can get away with registering under your social security number or whatever tax variant of EIN your country accepts for now.
Yes I mean legal registry, I am already running my store with Stripe linked to a Payoneer account and Paypal Express Checkout using an merchant account (unverified) without having any kind of legal registration but I may need to do it soon.