This is not possible programmatically, due to the potential for a script to misuse the blocking or filtering of errors in the console to hide its activities from the Chrome user.
You can of course filter messages in the console by Error, Warn, Debug.
You can add your voice to those asking for more powerful capabilities to filter console messages in Chrome, so when you view the console you can filter out the messages that clog it up. One of the posts does offer a tool to tag your own console messages and then filter them.
If you want to highlight an error in the console, you can add styles to it when logging in your own code. E.g the following will make the message stand out:
console.log("%cUser %s has %d points", "color:orange; background:blue; font-size: 16pt", userName, userPoints);