Ajax Custom Post Filter is not filtering

Question

How to make your AJAX filter works properly:

As I said in the comments, add the data-category attribute to the <a> having the CSS class js-filter-item in the foreach loop in your archive-projetos.php template:
```
// * Wrapped for brevity.
foreach ( $categories as $cat ) : ?>
    <li><a class="js-filter-item"
        href="<?= get_category_link( $cat->term_id ); ?>"
        data-category="<?= $cat->term_id; ?>"
    ><?= $cat->name; ?></a></li>
<?php endforeach; ?>
```
And if you wonder why add that attribute, then it’s because in filter.js, the script is reading the category ID via $(this).data('category') which is equivalent to $( this ).attr( 'data-category' ).

In the filter_ajax() function in functions.php, replace this:

$query = new WP_Query($args);

if(isset($category)) {
    $args['category__in'] = array($category);
}

with this:

if ( ! empty( $category ) ) {
    $args['tax_query'] = array(
        array(
            'taxonomy' => 'Tipos de Construção',
            'terms'    => $category,
        ),
    );
}

$query = new WP_Query( $args );

I.e. Switch the position (of the new WP_Query() call and the if block) and then use the tax_query parameter and not category__in which is for the default category taxonomy only.

Also in that filter_ajax() function, replace the $category = $_POST['projetos']; with $category = $_POST['category']; because your JS script is sending the category ID in a (POST) input named category and not projetos.

However, you should always check if an array item is actually set before attempting to use it, so you would want to use $category = $_POST['category'] ?? ''; or the old way (prior to PHP 7): $category = isset( $_POST['category'] ) ? $_POST['category'] : '';.

Additional Notes

Even though the Tipos de Construção does work as the taxonomy name, i.e. the first parameter you pass to register_taxonomy(), you should actually use a slug like tipos_de_construcao because the documentation says:

$taxonomy is the name of the taxonomy. Name should only contain lowercase letters and the underscore character, and not be more than
32 characters long (database structure restriction). Default: None

So you should try to fix the taxonomy name, i.e. use a slug. It will take time, but it will be a better option in the long run. =)