The presence of the ArcGISIndexingServer.exe service running in the Windows Task Manager is not an indication that indexing is in process. The service runs in the background independently. Many operations in ArcGIS Pro causes it to start. Once it is running, it remains running for the life of the ArcGIS Pro application until ArcGIS Pro is shut down and it finished its work. If the indexing server is stopped while it is running from the Task Manager while ArcGIS Pro is still running, additional operations in ArcGIS Pro causes it to start up, it starts again. This is expected behavior.
If the indexing process is actually in the process of indexing content that was added to a project and is shut down in the middle of its work before that work is complete, incomplete search results are returned.
This issue is resolved for ArcGIS Pro 2.4 to provide messages more consistently when indexing is just starting, and to progressively show search results whenever possible. That is, it is expected behavior if searching while indexing is in progress, some search results matches the search criteria. Searching again (by clicking in the Search control and press Enter, do not try to Refresh the search results) may get more results as indexing has indexed more items since the original search. A message should indicate that indexing is still in progress even if there are search results.
From the description, the software is behaving as expected.