It is confusing-sorry for this.
I do not have the official LR options for ~11€/months.
I have a official Lightroom CC mobile version with 200 GB cloud for ~5Euro/month that works only on Apple mobile devices-not on desktop computers. On my PC, I have a license for LR 6, that can be synchronized with my LR CC version.
Adobe cause most of the confusion... How many plans are out there?!
With LR6, is it logged into the same account that is used with LRCC Mobile? Maybe go to Help > System Info, and copy the first line. That will help make clear which version you are using. If LR6 will sync, enable sync in the catalogue (Preferences) and it should automatically download everything from the cloud. If LR6 will not sync, then the Lightroom Downloader will download everything that is in Adobe's cloud.