Are PostgreSQL clusters and servers the same thing?
tl;dr

Yes.

Confusion arises from the official terms used in the SQL standard versus more commonly used terminology. 


Cluster = database server
Catalog = a database
Schema = a namespace, a grouping of tables


Computer > Cluster(s) > Database(s)/Catalog(s) > Schema(s) > Table(s)

See my Answer to a similar question on Stack Overflow for more info.

Briefly here…

An installation of Postgres on a computer is called a cluster, per Section 4.13 of SQL-92 titled "Clusters of catalogs".

That cluster has one or more databases, each database called a catalog per the SQL standard. Each database contains one or more schemas, each schema being a namespace and security boundary for a collection of tables.

You can have more than one cluster on a computer. This is typically done for different versions of Postgres. For example, you could have Postgres versions 9.5, 9.6, and 10 all installed, each listening on a different port for incoming connections. That would be three clusters in that computer, each containing catalogs that contain schema that contain tables that contain your rows of data defined by columns.