Support for two factor authenticator apps

But wait, isn't the phone number uses two-step verification instead of two-factor authentication?

The site you linked states in the Q&A section „Two-factor authentication is built directly into iOS, macOS, tvOS, watchOS, and Apple’s web sites” suggesting this as a fundamental feature of the two-factor authentication and without these, we could not talk about two-factor authentications. A phone number is none of those, so which one is used then?

The answer for the question „What if I use two-factor authentication on a device running older software?” further implies that you cannot use it with just everything. Although technically „phone number” isn't everything, while also the answer only related to Apple product, the answer further raises concerns whether it will be able to use phone numbers for two-factor authentications.

Also doesn't a text message (SMS) or a phone call will be charged? Don't give me wrong, it's not that I'm not willing to pay for a service, it is just a matter of what am I actually paying for.