How to add day number and initial to my post graph?

I would go like this:

A. Use the Unix timestamp as array key.

$comment_counts[$date->format('U')] = $query->post_count;

B. Loop with key included.

foreach( $comment_counts as $count_key => $count ) :

Then you obtain the day number and the initial of the day from the key:

// "d" means with leading zero, use "j" in place of "d" for no leading zero 
$day_number = date("d", $count_key);
$day_initial = substr(date("D" , $count_key), 0, 1);