Hey there janssensn,
Thanks for reaching out to the Apple Support Communities.
Just to confirm, when you say that you can't use your Apple ID on mobile data, are you unable to use apps that require internet access when not connected to Wi-Fi?
If so, while we understand that you've enabled Mobile Data in Settings > Mobile, do you also have all the apps enabled (green tab showing) under the "Cellular Data" section.
If you do have all apps enabled, do you by chance see No Service or Searching in the iPhone status bar when not connected to a Wi-Fi network? We ask because this would indicate that your iPhone isn't connected to a cellular network.
For more details on troubleshooting issues with seeing No Service or Searching, check out this next link here: If you see No Service, Searching, or SOS on your iPhone or iPad - Apple Support
If you're not seeing No Service or Searching, and you're only having issues with your cellular network, it may also be helpful to reach out to your carrier to check for any issues or possible outages on their end.
Keep us updated on what you find, and we'll continue from there.
Have a great day!