Should I use my primary Apple ID for that or create another one to use with Apple Developer account?
From the Apple Developer Program page:


  Enrolling as an Individual
  
  If you are an individual or sole proprietor/single person business, get started by signing in with your Apple ID with two-factor authentication turned on. You’ll need to provide basic personal information, including your legal name and address.


If you use a different Apple ID, you need at least one device with this Apple ID signed-in to authenticate your login, because of mandatory two-factor authentication. Switching your Apple ID on a device is not something you want to do regularily.