I assume just setting the same SSID and password is insufficient.
That in fact should be enough, as long as the new AP is actually bridged to the same network. But also double-check the security mode (open/WPA/WPA2) – some devices completely refuse to automatically roam between APs which use different security levels.
(I've only used powerline devices from TP-Link, and that's the only mode they support. But it's possible that some devices can be switched between bridge and router modes, and it's the latter that would cause you problems here.)
(Quick check: connect to your "extender" network, and check whether your computer's ARP cache still shows the main router's MAC address. If the gateway IP is different, or if it has a different MAC, that means you've created different subnets and roaming will be troublesome.)
Note that even if you have everything configured correctly, devices might still be reluctant to roam between different APs unless the current signal is very poor. (Roaming causes a short interruption which might annoy VoIP users or online gaming players. More expensive APs have features like 802.11r or proprietary alternatives to avoid this delay.) Some Windows wireless drivers let you configure the roaming "sensitivity" or "aggressiveness". macOS prefers 5 GHz as long as it's usable.
I'm also curious about how cloning works.
"Cloning" simply makes the device act as a WPS client, and copies three parameters from the network that currently has WPS active: the network name (SSID), the security mode (WPA/WPA2), and the passphrase (or at least the hashed version called the PSK).
It does not do anything else – it's still up to you to physically connect everything.