We always want you to get the most out of your phone. I can confirm that we do use dynamic IP address that is always going to be different. These sites will be looking for the same IP address to identify you which is typical of a home based internet service. I apologize for any inconvenience this causes.
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In addition to the IP address information mentioned by Verizon Support above, some secure sites are set up to prevent automatic login from being possible. This is common across banks and other sensitive sites. Many will also not permit passwords to be automatically filled in, or the browser won't recognize the page as a login form.
Additionally, make sure you have Cookies enabled. Settings > Safari > Cookies should be set to "Always allow." Any option which turns off cookies or deletes cookies after the session has ended, will prevent automatic login.
Make sure Safari, Chrome, Firefox or any other browser you use on your phone/tablet is not set up in Private Browsing mode. The phone will not store any information on private browsing sessions. If the browser crashes or is closed, those tabs go with it.
Make sure the phone in Settings > Safari is configured to save passwords. Same deal with any other browser you use.