When you add a map service to ArcGIS Pro, you aren't getting the raw feature data from the server. As you pan and zoom around, requests are made to the server to generate images (typically PNGs), and those images are sent to Pro and displayed.
Depending on how the map service is configured, you may be able to add one of the sublayers to Pro on its own and get feature data. For this particular map service, you can do this, by adding this URL:
https://spatial.industry.nsw.gov.au/arcgis/rest/services/PUBLIC/StreamOrder/MapServer/0
There are restrictions on this approach, however. In the case of this particular server, you can only fetch 1000 features at a time, which might work for a small area, but is far less than the total feature count of almost 2.8 million records.
Although it might be possible to write a script to scrape this, your best bet is to look at the data owner's website for a download link, or contact them and request the data in its entirety.
Edit - Also want to mention that being able to extract a shapefile from a map service isn't a shortcoming in Pro functionality; in some cases, map services contain data that can't be represented in shapefiles (such as raster data), and in some cases a data owner may want to restrict access to the raw data, but still allow access via server-rendered images.
