Does Google's 2-factor authentication only care about the mobile phone number?

Nope, it only knows the phone number. That's the only information that it has. Just your phone number. Any facts about what's on the other end of that phone number are irrelevant, as far as Google is concerned. See my first answer for a link to the app on Android, though http://support.google.com/accounts/bin/answer.py?hl=en&answer=1066447 explains how to set it up on Android, iPhone, and Blackberry,

I've dealt with this many times. This is the cleanest way I've been able to do this, after flashing many custom ROMs and futzing with my phone. If anyone knows of a simpler way of doing this, I'm all ears. If you've got an Android device, this is how I'd do it when switching devices (again, this assumes you're using an Android device, using the Google Authenticator app, and a QR code scanner such as the free Barcode Scanner v4.2 by ZXing ): - go to http://www.google.com/accounts/SmsAuthConfig - enter your password, then turn off 2-step verification (top link) - verify 2-step auth has been turned off (you'll get an immediate e-mail confirming this) - under "How to receive codes"/"Mobile application," select Remove/Replace - on your new phone, go through the entire setup for Android-- you don't have to use Authenticator anymore as it's turned off - go back to the 2-step verification page at the above link - turn 2-step verification back on - add mobile application, verify phone number is correct (should be the same, as you're just replacing the phone) - a code will be sent via SMS, enter code to confirm - add mobile application for Android, which will present you with a QR code - go into Google Authenticator's menu, select "Add account," and "Scan barcode" - scan QR code, enter confirmation code on computer, and you're set. This doesn't take as long as it seems. I've done this numerous times. PM me if you need help with anything. Last note: having printable backup codes is a lifesaver. Use it! Keep them safe, but accessible!

(And the reason I turn off 2-step is because when setting up a new phone or starting from a newly flashed ROM, you avoid the mess of entering a backup code or using your old phone's Authenticator in setting up the new phone. It's a terrific pain in the ass. Better to just set up the new phone and add 2-step after it's all set up.)

herrdoktor's advice is good if you already were authenticating via an app instead of via a phone number -- and turning off Authenticator is another good idea if you're nervous, but since you're currently authenticating via a phone number, it will continue to work with a new phone without you doing anything.

I'm confused about the question. Is it whether they will give you the codes at all after you get a new phone? Google will send the code to the phone number you specified, it doesn't know what's on the other end of that. If you have a smartphone you can get the https://play.google.com/store/apps/details?id=com.google.android.apps.authenticator2&feature=nav_result#?t=W251bGwsMSwxLDMsImNvbS5nb29nbGUuYW5kcm9pZC5hcHBzLmF1dGhlbnRpY2F0b3IyIl0. (that's the android version, since you didn't say what type of phone you have, I know there's an iPhone version, don't know about other mobile OSes but I assume they exist too) that will generate the codes instead of texting or calling you. You can also print out http://support.google.com/accounts/bin/answer.py?hl=en&answer=1187538 that you keep in your wallet or whatever in case you don't have access to your phone when you need to sign in somewhere.

In case you're asking the other question that your phrasing can mean -- it doesn't associate authentication with a phone number, only with a device. Actually, even less than that -- if you are authenticated via Chrome on your computer, you are not also authenticated via Firefox, etc (and clearing cookies will clear your authentication). And different google apps sometimes need to be authenticated separately. So you will have to re-authenticate yourself on your new phone when you get a new phone even if you keep the phone number.

If you're asking about google calling your phone, you're fine - that part of authentication will still work. It's connected to the number, not the device, so as long as you can answer whatever phone is attached to the number, you're set. If you're asking about using your phone with your google account (gmail, google play, etc.), assuming that you're talking about an android phone, you will need to authenticate that device (I assume you use device-specific passwords, if you don't, consider it!) separately. You will be able to do that just as easily as the first time you signed into your device, from your computer or whatever.

Oh, sorry for the confusion... I have 2-factor enabled. The code is sent to a phone number. I want to change the handset associated with that number (including SIM card). Will getting a new mobile device cause problems? The mobile telephone number associated with the account is a key part of 2-factor authentication.

OK, then no, you're fine. I just got a new phone, swapped the SIM card, it started sending the messages to the new phone, no problem. It cares about the phone number, not the device. If you're at all nervous though, printing out a set of backup codes before you do anything will give you some peace of mind, and setting up the app on your new phone instead of having them call you will just be easier moving forward.