I would like to override chrome.storage.local.set in order to log the value of a storage key when the Chrome Storage API is called:
var storage_local         
        
You need to use .call() or .apply() to provide this object:
storage_local_set.apply(this, arguments);
The parameters in your override don't match the documentation though. If you want to match the documentation then do it like this:
const api = chrome.storage.local;
const { set } = api; 
api.set = function (data, callback) {
  console.log(data);
  set.apply(api, arguments);    
};
And if you want to change the signature to separate key and value:
const api = chrome.storage.local;
const { set } = api; 
api.set = (key, value, callback) => {
  console.log(key, value);
  set.call(api, {[key]: value}, callback);    
};