AppleID Two-factor Authentification with Email address Suppose I have 2FA activated for AppleID, and I lose my phone and I need to log in with my AppleID. This becomes further complicated when I travel to another country and have to use another, local phone number for some time.
I wish I could use my email-address for 2FA instead of my phone number.
Is there any way I can log in with my AppleID if I don't have access to my phone number?
Keep in mind SMS is a fallback from trusted device push notifications. You don’t even need an SMS message unless you can not / do not choose to have trusted devices with you that use network based Apple push notification services (APNS).

Go all in on trusted devices if you can - avoid SMS entirely.
Choose a dual sim phone if you can so you can layer both lines when you travel or add a short term number.
Many carriers allow SMS to forward to a watch or iPad or computer so that the delivery is network based like the Apple trusted device notification path.
Add your new number as a trusted phone number when you need that extra flexibility.


https://support.apple.com/en-us/HT204915

You don’t have to remove your primary or depend on SMS only would be my recommendation unless/until Apple adds new methods of verifying it’s you when you want to sign in to a service from abroad.