Hello eprebys,
Welcome to Apple Support Communities. We understand that Universal Clipboard is not working as expected on your Mac. We are happy to point you in the right direction.
To get started, we'd recommend ensuring that all devices are updated to the latest software version. You mentioned you're using an iPad and a MacBook Pro, however we wanted to ensure any other devices you may own are also updated. You can do that using the steps here, as this often resolves abnormal behavior such as this:
We'd also suggest toggling Handoff off, restart both devices, then toggle it on again. Handoff affects whether devices see each other, so we'd like to determine if the Mac is able to see the iPad using the Handoff feature after completing this step. While we know you came to us from this article, we wanted to ensure this step was completed for isolation purposes: Use Handoff to continue tasks on your other devices - Apple Support
If you're unable to see the iPad, we'd suggest restarting your modem and router, then trying again. Once complete, let us know if Universal Clipboard is working in other apps such as Notes, Mail, Safari, etc. This can help us isolate whether this is specific to a particular app, or software.
Please let us know if you have further questions.
Take care.