Umstellung von PS Elements 6 auf PS Elements 2018

There is a better solution than writing the tags to the files and importing. It's to convert the PSE6 catalog from the catalog manager in the new version. Check in your Explorer/Finder for the location of the catalog.psedb file which is the database in the PSE6 catalog folder. You should browse to this location to start the conversion, which should restore nearly everything from your old catalog, not only tags and captions, but also albums (collections), stacks and version sets.

As I understand it, the tags are only written to the Exif information when you choose to do so 

so select photos and press CTRL-W to write the tag information to the photos. 
if you have not done that in the old PSE version, I thing you cannot do that afterwards. Unless off. Course you still have got the old software installed and are able to write those tags to the exif info on the pictures