It's possible to have multiple Datadog orgs open in the same browser context by having an admin from your account enable custom subdomains (https://docs.datadoghq.com/account_management/multi_organiza...).

It would also help with links routing to the correct org:

> If you are a member of multiple organizations, custom sub-domains help you identify the source of an alert or notification. Also, they can immediately switch you to the organization associated with the sub-domain.