Hi @HeatherM_JDI,
> When publishing services to a federated ArcGIS Server, what is the difference between configuring the share settings via the Portal interface and using Server Manager?
In the case of a federated ArcGIS Server with an enterprise portal, there is no difference. The resultant behavior is the same.
> ... where we've noted that sharing via Server Manager is the required workflow, then it is very confusing to review sharing settings in Portal where it looks like the content is not shared at all.
Basically in the two screenshots you posted, ArcGIS Enterprise offers the ability to enable sharing of web services at the ArcGIS Server tier and at the enterprise portal tier. Apologies for the confusing UI/UX - this is likely a byproduct if how ArcGIS Enterprise has evolved over time. When Portal for ArcGIS (aka. the enterprise portal) was first released, it was optional to federate ArcGIS Server with it. WIth each Enterprise release, the Enterprise base deployment is now a "standard" deployment pattern, but the UI/UX is still available in both Enterprise components.
Hope this helps,