Hi Sorato,
In order to publish a mosaic dataset as a map service, you need to have an ArcGIS Server site licensed with the Image Server role.
As a workaround, you could export the mosaic dataset to a single raster dataset using the Mosaic to New Raster tool: Mosaic To New Raster (Data Management)—ArcGIS Pro | Documentation.
This way, you are publishing a single raster dataset as a map service, which does not require an additional license for ArcGIS Server.
Hope this helps!
-Calvin