Using New Outlook on Windows for professional communication and productivity
You indeed need to add a subscription if the account associated with the Exchange Online account doesn't have a license for the Office apps.
When adding a subscription, make sure you use your Microsoft Account. In the case that your Microsoft Account is associated with a 3rd party domain, it could lead to authentication issues. In that case, log in with an outlook.com alias for your account instead. If you don't have one, you can easily add it here;
https://account.live.com/names/Manage
You can check which aliases can be used for login here as well.
Another way to go would be to first configure New Outlook with your Outlook.com account, add the Outlook.com account as a subscription account, add the business account, and then remove the Outlook.com mailbox account.