How to catch all PHP errors with custom error handler?

Yes, there is a way to catch PHP errors (notice, warning, error) with a custom error handler by using set_error_handler():

$logger = new Monolog\Logger( 'PHP-ERROR' );
$logger->pushHandler(
    // Configure your Monolog handler here
);
set_error_handler( function( int $errNo, string $errStr, ... $args ) use ( $logger ) {
    $errorName = 'NOTICE';
    $errorLevel = Monolog\Logger::NOTICE;
    switch ( $errNo ) {
        case E_USER_WARNING : 
            $errorName = 'WARNING';
            $errorLevel = Monolog\Logger::WARNING;
            break;
        case E_USER_ERROR : 
            $errorName = 'ERROR';
            $errorLevel = Monolog\Logger::ERROR;
            break;
    }

    $errorMessage = "{$errorName} {$errStr}";
    $logger->addRecord( $errorLevel, $errorMessage );

    // return false if you want the default error handler to proceed
    return true;
) };

This is just an untested example but basically the way of how to send PHP errors to Monolog. Thus you can use the power of all the existing Monolog handlers.

If you want to log custom WordPress errors (like DB errors, HTTP errors, failed log ins) you have to listen to the specific hooks of the WP API and delegate messages about these events to Monolog.

However, if you are able to use third-party packages, I would suggest to have a look at Wonolog. A library that aims to connect WordPress to Monolog. It does exactly the same as above plus adds a comfortable API to plugin developers to log anything they like.

Edit: If it is only about PHP errors and Monolog, you can use the error handler shipped with Monolog instead:

$logger = new Monolog\Logger( 'PHP-ERROR' );
(new ErrorHandler( $logger ) )->registerErrorHandler()