The idiom “a cat’s paw” means a person or persons (or even a country) that is being used by another to achieve an objective and especially if that objective is being achieved in a cynical or duplicitous manner.
It’s interesting that China appears to have accused the Australian government of acting as a “cat’s paw” for the US. But normally it is used, as I understand it, in relation to individual people such as a person using another to stir up trouble with a third party.
Sadly, the idiom is a slight denigration of the domestic cat because it implies duplicitous behaviour. This reinforces a stereotype that the domestic cat is sly, tricky and unreliable. Some people do have misconceptions about the domestic cat often in relation to their character. Domestic cats are highly reliable, innocent creatures who wear their heart on their sleeve to use another metaphor. They are not sly and devious. I feel the need to defend the domestic cat as usual.
In the case of China and the US, I think the reference to a ‘cat’s paw’ is the fact that United States agriculture exports to China have boomed after Australia criticised China for their treatment of Muslims in their country. China reacted by restricting imports from Australia which in turn has resulted in enhanced exports from the US to China. That is my reading of the situation and how the idiom has been used in this instance.
An ‘idiom’ describes ‘a group of words established by usage as having a meaning not deducible from those of the individual words’ – Google. In other words, people use words that have no meaning in relation to the events described but through usage have come to encapsulate the events in a succinct phrase.
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