Hi Hayley,
If you are using the same portal admin account you can likely rule out a lot of issues tied to privileges or ownership.
Likewise, if the data was not registered with server you would get the option and then receive a warning that the data would be copied to the server (as a file geodatabase).
Using the same machine also rules out issues like different versions of Pro.
Are you able to expand upon what the data is? Can you confirm it is the same map and layer as the other user? If it is a database connection, how is it authenticated? Can you share a screenshot of the map content pane?
Any other differences you can identify between you and the other user who can publish ok?
Would probably recommend reaching out to Esri technical support who may be able to actively look into it. Most issues with publishing are tied to errors or warnings; not absence of options.
What do you have set as your active portal? Is it the Portal for ArcGIS or ArcGIS Online? You may need to switch active portals, even if you are signed into Portal for ArcGIS.
Cheers,