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.