urbinati ,
there recently has been an FAQ published but this in more detail isn't part of it yet.
Until then you can get more details at the Ubuntu Pro description which outlines the entitled coverage. It also goes into more detail about what is considered a VM, what a Node is and what isn't and how they will be accounted for in clusters of various kinds.
But that does not yet cover the "how do I then attach it" which you asked for. Trying to answer your question part by part.
How can I subscribe a EXSi node/host?
You'd buy Ubuntu Pro for that ESXi based server (which is a "node" in the terms of Ubuntu Pro description) and attach all Ubuntu guests to that contract.
Is there like a dedicated client?
There is the Ubuntu Pro Client which you'd use in your Ubuntu instances. There is not client (needed) for the VMware Host as the Host itself would not be covered as Ubuntu Pro does not provide updates for VMWare.
And what about Proxmox (qemu/KVM based)? Should I just subscribe the VMs ...
Yes - as with Vmware you'd subscribe all VMs indeed, the host is again the "node" in the Ubuntu Pro description terms. But also again the Host itself would not be covered as Ubuntu Pro does not provide updates for Proxmox.
... and the hosts gets automatically "subscribed"? How does it work actually?
There is no automatic guest->host or host->guest distribution of the attachment to Ubuntu pro that you suggested because that would break isolation.
Thank you for bringing it up, please take all of the above with a grain of salt until we have further clarified all the terms based on your and similar requests. That clarification is worked on via a similar case that someone else just recently filed as an issue against ubuntu.com. There where we will work on resolving this in more depth, until then I hope the answer above already helps.