Accounts on their Element Home plan can work on Android and iOS apps too. It's just that copying over data from a different homeserver to this new homeserver is possible only on the desktop app at the moment.
The ability to copy traffic history from one homeserver to another is a huge deal. I have not succeeded in discovering any details of this rollout. What server versions support it? Does it suffice to copy using one client, once, and then all clients see the result? Or, only after pointing those other clients at the new server? Do the other clients only see old traffic, after, until they are switched over, too?
How can I have more thsn one homeserver watched? Do I need to run N independent client apps (e.g. Element and Fluffychat), with each pointed at one homeserver?
It is very hard to find out any details of what is handled where in Matrix. Nobody in rooms devoted to services acknowledges questions, never mind answering any.
As you rightly say, you can use any client for Element Home and the only restriction is that the in-app setup and migration wizard is only available in the Element web clients at the moment (this is now corrected in the blog article).
That said, the Matrix->Matrix migration tools (https://ems.element.io/tools/matrix-migration) can be used outside of the setup flow, to help migrate your existing chats to new accounts. So people can always use those separately from host setup, if needed.
How can I have more thsn one homeserver watched? Do I need to run N independent client apps (e.g. Element and Fluffychat), with each pointed at one homeserver?
It is very hard to find out any details of what is handled where in Matrix. Nobody in rooms devoted to services acknowledges questions, never mind answering any.