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.
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.
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.