This issue was logged as BUG-000114029 in May 2018, and has been marked as "Not in Current Product Plan by Esri Development. However, effective with version 2.7 of ArcGIS Pro, it is possible to add data to the Pro project that does not have a projection defined. In earlier versions, a default coordinate system was assigned to the data - either GCS_WGS_1984, or WGS_1984_Web_Mercator_Auxiliary_Sphere - which were wrong in most cases.
The extent of the data which has the unknown projection definition can be viewed in the Catalog Pane in ArcGIS Pro, and with some preliminary research about coordinate extents for the area, in commonly used coordinate systems for the area, it is usually pretty easy to figure out what projection the data was created in. If nothing matches, i.e. CAD data, it may be necessary to create a custom projection file to align the data.
The fundamental issue to remember is that Project on the Fly works. If the data has a projection defined, and you have set the appropriate datum transformation, but the data does not show up in the correct location, the assigned projection is wrong. Clear that projection definition, and follow instructions in Knowledge article 000007880 and related articles to identify the coordinate system, or create a custom projection to align the data.