You can integrate a single Salesforce org with multiple Genesys Cloud orgs by creating a separate Lightning app for each respective org in Genesys Cloud regardless of whether the CX Cloud, Voice for Salesforce Service Cloud Voice package is installed. 

Follow the steps to set up the CX Cloud, Digital and AI for Salesforce Service Cloud Voice and in the final step to create a Lightning app, ensure that you select the following details:

• Add the utility items – Omni-Channel, Genesys Cloud for CX Utility and CX Cloud, Digital and AI.
• Add the navigation items – Experience

In this approach, the combination of the Omni Channel widget and other utility items configured within the Lightning app works for multi-org. When you use Salesforce Service Cloud Voice, an agent can belong only to a single contact center in Salesforce and their assigned contact center represents a CX Cloud from Genesys and Salesforce integration. When agents use the Lightning app, they are automatically associated with the corresponding Genesys Cloud org. You can create Lightning apps for any number of Genesys Cloud orgs.

When Salesforce Service Cloud Voice is not used, the agents do not belong to a contact center and the Genesys Cloud org is determined based on the CX Cloud, Digital and AI utility item configuration. Similarly, you can continue to add a new Lightning app as required when Salesforce Service Cloud Voice is not used to accommodate the integration to additional Genesys Cloud orgs.

For more information, see About CX Cloud from Genesys and Salesforce, Set up CX Cloud, Digital and AI for Salesforce Service Cloud, and Create a Lightning app for CX Cloud, Digital and AI for Salesforce Service Cloud.