How to check if a post is published today?

You can use get_the_time() to get the date of the current post. Just use a format that only includes the year and date, and compare it to the current date using the current_time() function with the same format:

if ( get_the_time( 'Yd' ) === current_time( 'Yd' ) ) {
    // Post was published today.
}