Hi, waldhaus!
Thanks for asking about managing your Messages here in the Apple Support Communities. It looks like you want to be able to delete a message or thread from one device, like your iPhone, and have it also delete from other devices, specifically your Mac. We're happy to help!
If you're using iMessage and SMS forwarding, this allows your message to be received on other devices from your iPhone. It does not sync deletion across device, however. For this, Messages in iCloud needs to be enabled on all devices: Keep all your messages in iCloud - Apple Support.
If you've already got Messages in iCloud enabled but are still not seeing management of your messages sync to other devices, you can try the following steps:
- Restart the iPhone: Restart your iPhone - Apple Support and the Mac: How to turn your Mac on or off - Apple Support.
- Turn off Messages in iCloud on all devices using it, then re-enable it -- starting with your iPhone.
- Check for, and install, any available updates for your Mac: How to update the software on your Mac - Apple Support and your iPhone: Update your iPhone, iPad, or iPod touch - Apple Support.
Cheers.