This is a question that coimes up all the time in entertainment technologies.
Every creditable set of regulations worldwide requires that the flexible cord must be protected from damage due to excessive current flowing.
The equipment mounted circuit breaker will ensure this under normal operating circumstances. So having a 60A socket with a 60A plug in it with a 15A rated cable to a machine protected with a 15A breaker is a safe situation; there will never be more than 15A flowing through the cable. However, not all juristictions see it this way, some allow, some do not.
The other issue is under short circuit condiitions where you cannot rely on the 15A breaker to protect the cable, the fault may occur prior to the breaker, for example, in the cable itself. Then the question becomes will the cable be (further) damaged by the shoprt circuit current that will flow until the upstream breaker opens? There are many variables involved in that question.
So in practice, if you have a threre foot cable that has an oversized plug on it and a correctly rated breaker in the thing, generally that is viewed as acceptable. A fifty foot cable wouldn't be and would require better protection.