Many organization integrate applications with shared databases

Shared database integration has been encouraged by presence of SQL as a standard query language.
It couples applications to each other by sharing database structure, making it harder for applications to change rapidly.
It encourages a single database technology and schema for all applications, making it harder to use the appropriate database technology for a single application's needs.
This makes reporting easier for simple cases as reporting tools are plentiful for SQL. But reporting needs often slow down applications and can only report on data within the shared database.

instead encapsulate databases and share via service APIs

An application database is used only by a single application. Any external integration is done through APIs built and exposed by that application.
By encapsulating a database through an application API, clients are no longer directly coupled to the database technology and structure.
Application APIs provide a greater range of data models than the underlying database model.
A problem is that analytics clients may need APIs to be specially created for them to get at important data in an effective manner. If this is not important to the application development team, there can be frustrating delays and handoffs. The Service Custodian approach can help with this.