Very simply, the answer is: No, Ford does not use a Toyota hybrid system on the Kuga (known as the Escape here in North America), although it is similar to Toyota's 2-motor hybrid system.
The hybrid system in the larger, RWD Explorer is a much different, 1-motor hybrid system. It is similar to the hybrid systems used by Mercedes-Benz, BMW and VW Group, sandwiching the single electric motor between the engine and the normal 10-speed automatic transmission.
I am not familiar with the Ford Puma, but from what I read, it uses a 48-volt mild hybrid system. It replaces the starter with a more robust starter/generator, with a idle-stop system and using regenerative braking to recharge the battery.
The Ford Kuga's / Escape's hybrid system uses a power-split device (PSD) with 2 motor/generators, 1 driving the wheels and the other as a generator. The conceptual design of Ford's PSD is very similar to Toyota's PSD.
This is where it gets complicated.
When Ford first developed its hybrid system and used it in the 1st-generation Ford Escape in the mid-2000s, it looked even more similar to Toyota's hybrid system, although both systems were developed independently. Ford even licenced Toyota's hybrid system at that time, to prevent Toyota from suing Ford for patent infringement, Ford said. Ford even had its PSD supplied by Aisin (Toyota's transmission partner/supplier); Ford claimed at that time that no other supplier other than Aisin knew how to build PSDs.
Ford has since then further developed its hybrid system independently from Toyota (so that both systems are more different now) and Ford has also started building its own PSDs (no longer buying PSDs from Aisin).
