Kind reminder: Kickstart is not a restoration script
Kickstart is the backup archive extraction tool. It only extracts the backup archive. It has nothing to do with the database restoration and the site reconfiguration. That is handled by the restoration script which was included in your backup archive at backup time.
This information is printed on your screen every time you run Kickstart, explained in our video tutorials, and included in several documentation pages.
Akeeba Backup's restoration script has allowed you to change the Cookie Domain and Cookie Path since August 2018. As documented, you can find this in the Site Setup page towards the end of the restoration process. Please make sure to pay attention to the big, yellow warning box on that page.
There is a good reason why this is not done automatically. The only time you need to provide a Cookie Domain and/or Cookie Path in Joomla is when determining this information through PHP and/or the configured Live Site parameter is not going to be correct. These options are manual overrides. By their nature of being manual overrides, it is impossible for any PHP software – be it Joomla or the restoration script – to determine their value automatically. That's why these are restoration options, not something changed automatically.
Therefore, there are two alternative solutions to your problem.
Option 1 (recommended): Leave Cookie Domain and Cookie Path empty in Joomla!. This is what 99.9% of sites need.
Option 2: Change the Cookie Domain and/or Cookie Path at restoration time in the Site Setup page to match the restoration target.
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!