How can I override print_embed_sharing_dialog() in WordPress 4.5

You can’t modify the core function’s output, but you can replace it with your own function by unhooking it from embed_footer and adding your own function with custom output:

remove_action( 'embed_footer', 'print_embed_sharing_dialog' );
add_action( 'embed_footer', 'my_custom_sharing_dialog', 9 );

function my_custom_sharing_dialog() {
    // write your own dialog html here
}

(I added it back with a priority of 9 to insure that it gets printed before the print_embed_scripts.)