https://developer.apple.com/apple-id/
Your Apple ID currently has two-step verification turned on, but
two-factor authentication is required. After you turn on two-factor
authentication, signing in to your developer account will require both
your password and access to your trusted devices or trusted phone
number.
How to make this 2-step auth?..
I was trying to login, and it didn't ask me to put SMS code or something else.
It means, "signing in to your developer account will require both
your password and access to your trusted devices or trusted phone
number" never happened.
In the same time it does ask 2-step auth on https://appleid.apple.com/.
What's wrong?
Best Answer
Two-step authentication is an older method of authentication for your Apple-ID. This is no longer "good enough" to be used with developer accounts.
You MUST upgrade to two-factor authentication as a developer. There's no way to login as a develop with two-step authentication anymore.
Apple has a detailed guide on enabling two-factor authentication here:
https://support.apple.com/en-us/HT204915
Basically you take your iOS device, open Settings and click your name at the top. Choose "Password & Security" and then tap "Turn On Two-Factor Authentication". You'll then be guided through how to verify your phone number.
If you want to do the same from a Mac, you have to open System Preferences and click "Apple ID". Choose "Password & Security" from the menu on the left and then click the "Turn On..." button alongside "Two-Factor Authentication".