Hi iknoerle,
Just to clarify, are you trying to set up iCloud contacts for the first time on your devices? If so, please check out this link, which outlines the correct process, and may be helpful: Set up and use iCloud Contacts
Set up iCloud Contacts
When you turn on iCloud Contacts, all of the information* stored on separate devices is uploaded to iCloud. If you turn off iCloud Contacts, your information won't move automatically from iCloud back to your device.
On your iPhone, iPad or iPod touch
1. Go to Settings > [your name] > iCloud.
2. Turn on Contacts.
3. When you're asked if you want to Merge or Cancel, tap Merge.
All of the information stored in Contacts is uploaded to iCloud.
On your Mac
1. From the menu bar at the top of your computer screen, choose Apple menu > System Preferences, then click Apple ID. If you’re using macOS Mojave or earlier, you don’t need to click Apple ID.
2. Click iCloud.
3. Select Contacts.
All of the information* stored in Contacts is uploaded to iCloud.
* If you have contact information stored in On My Mac, that information will not sync automatically. In Contacts, drag a contact from On My Mac to All iCloud, or export your contacts as Vcards, then sync manually with the Finder.
If your contacts aren't syncing after going through the above process, please be sure to review all of the information outlined in the link you came from, which explains what to do next: If your iCloud Contacts, Calendars, or Reminders won’t sync
Thanks!