We can use shortcode function to display post with category filter option. You have to add this function to functions.php . Here is the demo code

function posts_with_category_filter_shortcode($atts) {
    // (you can customize default values here)
    $atts = shortcode_atts(array(
        'posts_per_page' => 4, // Number of posts to display
    ), $atts, 'category_filter_posts');

    // Get categories for the filter dropdown
    $categories = get_categories();
    
    // Get selected category from the URL parameter (if available)
    $category_filter = isset($_GET['category']) ? $_GET['category'] : '';

    // Query posts based on selected category
    $args = array(
        'posts_per_page' => $atts['posts_per_page'],
        'orderby' => 'date',
        'order' => 'DESC',
	'cat' => $category_filter,
    );

    if (!empty($category_filter)) {
        $args['category'] = $category_filter;
    }

    $query = new WP_Query($args);

    // Start the output buffer to capture HTML content
    ob_start();

    // Display filter form (category dropdown)
    echo '<form method="GET" action="">';
    echo '<select name="category" onchange="this.form.submit()">';
    echo '<option value="">All Categories</option>';
    foreach ($categories as $category) {
        $selected = ($category->term_id == $category_filter) ? 'selected' : '';
        echo '<option value="' . $category->term_id . '" ' . $selected . '>' . $category->name . '</option>';
    }
    echo '</select>';
    echo '</form>';

    // Display posts
    if ($query->have_posts()) {
        while ($query->have_posts()) {
            $query->the_post();
            echo '<div class="post-item">';
            echo '<h2>' . get_the_title() . '</h2>';
            echo '<p>' . get_the_excerpt() . '</p>';
            echo '<a href="' . get_permalink() . '">Read More</a>';
            echo '</div>';
        }
    } else {
        echo '<p>No posts found in this category.</p>';
    }

    wp_reset_postdata();

    // Return the captured HTML content
    return ob_get_clean();
}

add_shortcode('category_filter_posts', 'posts_with_category_filter_shortcode');

Use this shortcode [category_filter_posts] for display the post and category filter option in your page.