Currently I have an Active Directory associated with a set of subcriptions for the company I work at\'s Azure platform. However, the \'Default Directory\' AD is on the domain
Q: "Does renaming the directory from 'Default Directory' possibly break anything?"
You can modify the directory name of Azure AD based on your requirements. It's just a display name, which will show up on the login page. The name modification will not break any links.
Q: "Does changing the domain name have any possible unintended consequences?"
The initial domain name was established when the directory was created by the admin, and it can't be changed or deleted. However, you can add a custom domain name.
Please see the referred part from official documentation below.
Every domain name in Azure AD is either an initial domain name, or a custom domain name.
Every Azure AD comes with an initial domain name in the form contoso.onmicrosoft.com. This third level domain name, in this example “contoso.onmicrosoft.com,” was established when the directory was created, typically by the admin who created the directory. The initial domain name for a directory can't be changed or deleted. The initial domain name, while fully functional, is intended primarily to be used as a bootstrapping mechanism until a custom domain name is verified.