How to get current post category details inside "loop"?

get_the_category works within the loop as well, don’t pass a post ID and it will default to the current post in the loop.

The first example in the docs for get_the_category show how to output the url and name of the first (or only) category:

$categories = get_the_category();
if ( ! empty( $categories ) ) {
    echo '<a href="' . esc_url( get_category_link( $categories[0]->term_id ) ) . '">' . esc_html( $categories[0]->name ) . '</a>';
}

EDIT: Your original code with the above example added:

<div class="container body-margin">
  <?php if ( have_posts() ) : while ( have_posts() ) : the_post(); ?>
    <div class="card">
        <div class="row">
            <?php $url = wp_get_attachment_url( get_post_thumbnail_id(get_the_ID()), 'thumbnail' ); ?>
            <img src="https://wordpress.stackexchange.com/questions/327572/<?php echo $url ?>" class="img-responsive card-image" />
        </div>

        <div class="row card-body">
            <?php

            $categories = get_the_category();
            if ( ! empty( $categories ) ) {
                echo '<div><a class="card-body-category" href="' . esc_url( get_category_link( $categories[0]->term_id ) ) . '">' . esc_html( $categories[0]->name ) . '</a></div>';
            }

            ?>
            <div><h4 class="card-body-title"><b><?php the_title(); ?></b></h4></div>
            <div><p class="card-body-date"><?php the_date(); ?></p></div>
        </div>
    </div>
  <?php endwhile; ?>
</div>

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