JS has no stdout or stderr - it just has the console.

It is possible to override console.log() with your own function that does whatever processing you want, and ideally then pass the log message to the original console.log function. Ditto for console.error.

For example:

(function() {
    const log_orig = console.log;

    console.log = function() {
        log_orig.call(console, 'log called with ' arguments.length + ' parameters');
        log_orig.apply(console, arguments);
    }
})();  // IIFE

You can't.

Any output and errors that browser outputs are not exposed to JS.