No, ArcGIS Business Analyst continues to be enhanced and improved with new capabilities and data updates. For using Business Analyst with ArcGIS Online you can either sign up for a trial or buy a premium app license. more info here:
I am very curious to know what made you think about its replacement.
In addition to what Praveen Srivastava mentioned, the current release of ArcGIS Pro introduced Business Analyst:
Business Analyst Comes to ArcGIS Pro! | ArcGIS Blog . It means that Esri considers that Business Analyst offers important analytical functionality and you can expect the functionality to be expanded, not only in Desktop, but also in your portal.
So, I know some of the extensions have changed names over the years. I guess that was my real question, is the name changing or is the Business Analyst name here to stay.