Is it possible to create a new Apple Developer account and create new Apple ID at same time?
Is the goal simply to have another account you can use to update the app in App Store Connect? If so, you do not need a developer account. All you need is for the owner of the developer account to invite you as a user and delegate access. Note that the invited account does not even have to be an Apple ID. This will grant you access to App Store Connect, but not all the resources of a full developer account. There are several roles and levels of permissions as detailed here: https://developer.apple.com/support/roles/

If you do need your own developer account, you must create the new Apple ID first, then enroll it in the developer program. You will have to sign into the new Apple ID from your iPhone initially to set a "trusted device" and enable two-factor (multiple Apple IDs can use the same trusted device). After you have set up your "trusted phone number" you can sign out from the new Apple ID and just use the SMS method for that account.