Hello Daniel,
Thank you for reaching out to us and I apologize for any inconvenience caused.
To update or merge your Rank Math accounts, you will need to contact our support team. Please navigate to the Rank Math Support Forum and open a ticket. In the Ticket Title section, provide a one-line summary of the issue, and in the Ticket Question section, provide more details about the email address update or account merger request.
In the Sensitive Data section of the ticket, enter the email addresses you want to connect to your PRO account. If you are merging your FREE and PRO accounts, please select one of the emails as your preferred email address. Additionally, if you have the invoice screenshot from FastSpring.com, please include it as well.
Once you have provided all the necessary details, click the SUBMIT YOUR TICKET button.
Here is an example of how the ticket submission page looks like:

Our support team will then locate your PRO account and proceed with changing the email connected to it or merging it with your FREE account, as per your request.
If you have any further questions or need assistance with anything else, please feel free to let us know.
Thank you for your understanding.
Best regards,
Rank Math Support
Sources:
I Purchased Rank Math PRO but Rank Math is Still Showing FREE » Rank Math
How to Fix PCLZIP_ERR_BAD_FORMAT (-10): Unable to find End of Central Dir Record signature » Rank Math
How to Change or Update the Payment Method » Rank Math
This is an automated reply from Rank Math Support Bot. If you need further assistance, please reply to this topic and a human support agent will reply to you shortly.