If you use the default automation to request a satisfaction rating from your customers, then follow-up tickets will also send the satisfaction requests if the automation conditions are met.
The condition Ticket Satisfaction | Is | Offered is reset to Unoffered when the follow-up ticket is created. This allows the automation to run on the new ticket just as it did on the original.
If you want to prevent the automation from firing on follow-up tickets, add the condition Ticket: Channel | Is not | Closed ticket to the Meet all of the following conditions part of your CSAT automation.
For more information, see the article: About automations and how they work.