How to add CSS class to custom logo?

You can use get_custom_logo_image_attributes filter

add_filter( 'get_custom_logo_image_attributes', function(  
$custom_logo_attr, $custom_logo_id, $blog_id  )
{
    $custom_logo_attr['class'] = 'your-custom-class';

    return $custom_logo_attr;
} ,10,3);

Just for anyone else that's looking for solutions. I found this, which I find much clearer than the accepted answer.

Plus it gives simple ways to change the URL on the link as well! Just a little more detailed than the accepted answer.

add_filter( 'get_custom_logo', 'add_custom_logo_url' );
function add_custom_logo_url() {
    $custom_logo_id = get_theme_mod( 'custom_logo' );
    $html = sprintf( '<a href="%1$s" class="custom-logo-link" rel="home" itemprop="url">%2$s</a>',
            esc_url( 'www.somewhere.com' ),
            wp_get_attachment_image( $custom_logo_id, 'full', false, array(
                'class'    => 'custom-logo',
            ) )
        );
    return $html;   
} 

WordPress provide a filter hook to custom logo customization. The hook get_custom_logo is the filter. To change logo class, this code may help you.

add_filter( 'get_custom_logo', 'change_logo_class' );


function change_logo_class( $html ) {

    $html = str_replace( 'custom-logo', 'your-custom-class', $html );
    $html = str_replace( 'custom-logo-link', 'your-custom-class', $html );

    return $html;
}

Reference: How to change wordpress custom logo and logo link class

I think I found one answer. But I really wonder if this is the right way? It feels a little bit dirty somehow: I simply copied the logo related parts from wp-includes/general-template.php into my theme's functions.php and renamed the functions with some custom classes added:

function FOOBAR_get_custom_logo( $blog_id = 0 ) {
    $html = '';

    if ( is_multisite() && (int) $blog_id !== get_current_blog_id() ) {
        switch_to_blog( $blog_id );
    }

    $custom_logo_id = get_theme_mod( 'custom_logo' );

    if ( $custom_logo_id ) {
        $html = sprintf( '<a href="%1$s" class="custom-logo-link" rel="home" itemprop="url">%2$s</a>',
            esc_url( home_url( '/' ) ),
            wp_get_attachment_image( $custom_logo_id, 'full', false, array(
                'class'    => 'custom-logo FOO-BAR FOO BAR', // added classes here
                'itemprop' => 'logo',
            ) )
        );
    }

    elseif ( is_customize_preview() ) {
        $html = sprintf( '<a href="%1$s" class="custom-logo-link" style="display:none;"><img class="custom-logo"/></a>',
            esc_url( home_url( '/' ) )
        );
    }

    if ( is_multisite() && ms_is_switched() ) {
        restore_current_blog();
    }

    return apply_filters( 'FOOBAR_get_custom_logo', $html );
}

function FOOBAR_the_custom_logo( $blog_id = 0 ) {
    echo FOOBAR_get_custom_logo( $blog_id );
}

Here's one suggestion how we might try to add classes through the wp_get_attachment_image_attributes filter (untested):

add_filter( 'wp_get_attachment_image_attributes', function( $attr )
{
    if( isset( $attr['class'] )  && 'custom-logo' === $attr['class'] )
        $attr['class'] = 'custom-logo foo-bar foo bar';

    return $attr;
} );

where you adjust the classes to your needs.

As you found yourself the_custom_logo relies on get_custom_logo, which itself calls wp_get_attachment_image to add the custom-logo class. The latter function has a filter, wp_get_attachment_image_attributes which you can use to manipulate the image attributes.

So what you could do is build a filter that checks if the custom-logo class is there and if yes add more classes.