Display Posts from Custom Post Type with Category Filters

Lifted straight from the codex:

<?php 
if ( in_category( 'pachyderms' )) {
    // They have long trunks...
} elseif ( in_category( array( 'Tropical Birds', 'small-mammals' ) )) {
    // They are warm-blooded...
} else {
    // & c.
}
?>

So, just use in_category to decide how to output each post – whether or not to make it a link, and what classes to use.

while ( $items->have_posts() ) :
    $items->the_post();

$in_category = in_category( 'identity' ); // this should be the slug for your category

Throughout the rest of your code you can then just check if $in_category is true or false, and if it is false skip the anchor tags, or set a specific class etc.