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Vaibhab ShahVaibhab Shah 

Adding custom tab in Service Console ??

Is it possible to add a custom lightning tag in a Service Console?On clicking the tab, custom lightning component will be shown.
However,workspace object will be on Case.

AbhishekAbhishek (Salesforce Developers) 
It's not possible right now.

Because of the Customization Limitations

To edit a Salesforce Classic console app, you must be in Salesforce Classic. From Setup, enter Apps in the Quick Find box, then select Apps and click Edit next to your console app.

You can’t upgrade a Salesforce Classic console app to Lightning Experience. You can choose to display or hide the app in the Lightning Experience App Launcher, but you can’t edit the app from the App Manager page in Lightning Experience Setup. To get started in Lightning Experience, customize these Salesforce-provided Lightning console apps: Service Console and Sales Console. You can also recreate your Salesforce Classic console app in Lightning Experience, but using Salesforce’s out-of-the-box app is faster and easier.

For further reference, you can check this,

But you a raise Idea in Idea exchange for it, Based on the Votes it will be implemented in the future.

I hope you find the above information is helpful. If it does, please mark as Best Answer to help others too.