Hi @frederik venø Thank you for posting your query on Microsoft Q&A.
The "Name" field within "Tenant Properties" here is a display name only - https://portal.azure.com/#view/Microsoft_AAD_IAM/ActiveDirectoryMenuBlade/~/Properties
Changing this would impact what your users see as the company name within Office and Teams etc.. but not any links for OneDrive/SharePoint etc.. as these should be using your tenant default domain name, which is changed through a different process and would cause links etc.. to change.
This process is outlined here - https://learn.microsoft.com/en-us/sharepoint/change-your-sharepoint-domain-name
Do let me know if you have any further questions, I would be happy to help!
Please "Accept the answer" if the information helped you. This will help us and others in the community as well.