While I don't know exactly what method you used to georeference, it sounds like you have a georeferenced image and CAD data with no defined projection. The steps you performed and the order in which you did it is going to dictate the outcome.
If you added the image and cad data to a map document, then your dataframe should have been set to projected NAD83 (based on supplied information). Order does not matter because it takes the first projected data added to determine the projection to use, and since your cad data doesn't have one it would use the image. If your cad data did have a projection, then the order you add the cad and image would matter.
Once you had both loaded, you georeferenced the cad to the image by whatever means. You do not state what format the cad data was in, but you say it has attributes, so I assume some sort of conversion was done. At this point you exported it to a shapefile or feature class which you then imported to a gdb. gdb's don't have projections - they are containers for feature classes that do have projections (among other things). When you exported the data, you should have been given a choice of which projection to use - dataframe, source, or another layer. It sounds like you chose source, which is the cad data and since that is unprojected so is your exported data. You could have chosen either dataframe or another layer and picked the image.
Since you didn't, you now have data that you've moved to the right place, but you haven't defined where that right place is. You need to use the Define Projection tool to assign all of your exported feature classes the same projection/coordinate system that your image is using. This is one of the few times you would actually use that tool - you know what it should be, but it's not set.
And to answer your actual question mark up there, yes it (feature classes in the gbd, not the gdb itself) is georeferenced, but not because it's based on the cad data; rather because it's based your modification of the cad data to be georeferenced. However you only saved the movement and not the description about the movement in your exported data. Hopefully that makes sense.