That is only true with dial-up, whether it's AOL, EarthLink, Level 3, or any of the others. With dial-up, your IP may change several times during one session. If you have cable internet, your IP remains constant. It may occasionally change, but that's rare. One way would be if you've had trouble with your service and your internet provider needs to resend a signal to your modem. Of course, if you change internet providers or move, you'll get a new IP.
Edit to add: While an IP is the surest way to determine if someone is a "sockpuppet", there are definitely other ways.