@patrykmaryn
Ideally if the credentials have been initialized once, they should be available to all the developers post that. However, since it's 403 which is related to Authorization, please check if you've been removed from any of the required profiles
Ensure you are a member of the Cloud Manager - Developer IMS Product Profile (which grants access to AEM Developer Console)
Sandbox AEM as a Cloud Service environments only require membership in either the AEM Administrators or AEM Users Product Profile