Wow, ok hopefully I am unpacking this question correctly. But let’s start with the question from the title.
Does Google et al. assume it’s your number or just a number you have access to? It’s the former. Google assumes you are entering your number. If you put in a communal number, that’s on you for screwing up the base assumption underpinning SMS as a second factor for authentication. When working with a factor which is supposed to be “something you have” it needs to be something that you control. Think of it like the keys to your home. If you aren’t the only person with a copy of that key, then that lock does not provide security for your home against others with the key.

As for the “DNA” question. I’m going to guess this is about websites “remembering” you for login purposes. The way this usually works is that, after the first login, the website sets a cookie in your browser. This cookie contains a cryptographic value which is also stored on the web server. When you go back to the site, your browser uses this value with your request for the site. The server then compares it to the stored value. If it matches, you are logged in, without needing to reauthenticate. It’s more complex than just sending the value, but that’s not worth getting into.

If you have multiple logins “remembered” this way, it may be possible to move to different accounts without the need to reauthenticate. Also, many modern browsers can save passwords for you. This lets the browser auto-fill your credentials for you. It’s universally a bad idea to save your passwords this way, but it could allow you to switch accounts without knowing the passwords.