I don't know what is this geoserver, but this port 8080 I think it's not relevant in this.
I might be able to help: http://geoserver.org/display/GEOS/Welcome. GeoServer is a Java web application that retrieves GIS data (as in PostGIS in the realm of PostgreSQL) and servers it up in standard formats. This allows for greater interoperability than would be possible with only the PostgreSQL/PostGIS database since many clients cannot communicate with PostgreSQL directly.
So the desired request chain is most likely something like this:
Client (E.g., ArcMap, a web page with OpenLayers or Leaflet, another web application that processes the data somehow) --> GeoServer (running on port 8080) --> pgBouncer --> PostgreSQL (with PostGIS extension)
It looks like GeoServer is having a problem connecting to PostgreSQL, so I think you're right. It's either between GeoServer and pgBouncer or pgBouncer and PostgreSQL.
Hopefully, that clears up some points of confusion.
