The reason airbnb, google and other companies ask you to verify your account is often due to geolocation, i.e. you will be asked to verify your account if previously you normally used a service from, say, London, and now you're using it from Vancouver.
I avoid this situation when faced with annoying service providers (particularly google), by using a VPN when accessing them. Since I'm always using the same VPN with the same apparent location (my VPN provider lets me choose), the service sees me in the same place and doesn't request reauthentication.
Traditionally a VPN (Virtual Private Network):
enables users to send and receive data across shared or public
networks as if their computing devices were directly connected to the
However, VPNs are also often used to bypass geographical restrictions such as using google in China, or a video streaming service outside of your home country. They're also used to protect your browsing and other communications from snooping or manipulation on an untrusted network such as a hotel or coffee shop.
I've also travelled to places where I didn't want to, or wasn't allowed to take my phone. I would leave the phone with a family member and inform them I might need them to read SMS messages to me. A slight alternative would be to change the phone number to someone you trust that isn't travelling and ask them to forward messages to you.
Services also store tokens in smart phone applications that allow them to recognise that it is being used by the normal user, and only if the application is installed on a new phone, would you be asked to authenticate. In this case, carrying your original phone with the apps installed, may evade the requirement to reauthenticate.
SMS authentication is likely to be phased out in the medium term as it is often seen as insecure.