After renaming of my MSSQL database instance I can't see it in the Management Console

Make sure you are checking the correct AWS region

If you don't know the correct region, you can check Billing (Cost Explorer). Dimension by Region, Filter - Service - RDS and see in which regions you have RDS

Hello.

I don't think RDS will disappear unless you delete it yourself.
Perhaps you have another region open?
Find your RDS instance while changing the region using the steps in the document below.
https://docs.aws.amazon.com/awsconsolehelpdocs/latest/gsg/select-region.html

If you can't find it using the steps above, check the event history from CloudTrail to see if there is an event that appears to be deleting an RDS instance.
https://docs.aws.amazon.com/awscloudtrail/latest/userguide/view-cloudtrail-events.html

Thank you for the answer to everybody, the region suggestion was perfect. I had to simply change the region to the correct one. Gabor