How ideologically similar are the Russian and Chinese governments?

There is no need for a monarch in the political system. This is a factor shared among America, China, France, and Russia.

Sovereignty does not belong to Allah. This is a characteristic shared among all non-Muslim countries.

The similarity between Russia and China is that Russia influenced China a lot during the modern Chinese revolution. The influence began with Mikhail Borodin. From Mikhail Borodin, till Mao broke away from Nikita Khrushchev， CPC was part of Communist International. The economic system of China before Reform and Open was the same as that of the Soviets.