Your code is fine, but your logic isn’t working because of $cat. $cat is returning an Array with an Object which you’re trying to check against. But you can easily just use in_category(‘music’) like this:
function add_go_back_btn($content) {
global $post;
$cat = get_the_category();
$post_type = get_post_type();
if(in_category('music') && $post_type === 'post') {
return '<div><a href="https://wordpress.stackexchange.com/music">This is a button</a></div>' . $content;
} else return $content;
}
add_filter( 'the_content', 'add_go_back_btn' );