How is the biological accuracy of ANNs typically measured?
One way the biological plausibility of an artificial neural network could be assessed is to look at how much a neural network abstracts away from the behavior of real neurons.

For instance, it is common in psychology and machine learning to use a sigmoidal activation function to determine the output of a node. If biological plausibility is a concern, one might prefer to use the Hodgkin-Huxley model. (See also: other neuronal models).

To many practitioners, biological plausibility is not a primary concern. In most ANNs, each node is not really believed to be implemented by a single neuron in the brain, and thus using a more realistic activation function does not necessarily make the network more biologically realistic.

Personally, I don't think looking at the behavioral accuracy of a biological neural network (compared  to an artificial neural network) says anything about the network's biological plausibility. Using more abstractions might produce more similar behavior due to less overfitting. Another concern is that the ANN could produce the same behavior as a BNN even if the actual computations involved are quite different.