How can I make the same settings for "EPT Tiles Item"?

Hi zub00, you can add field EPT Settings for EPT Tiles Item:
/admin/structure/paragraphs_type/ept_tiles_item/fields

Then you will need to update template for ept_tiles_item paragraph type.
Update classes for EPT Tiles Item:

{%
  set classes = [
  'block',
  'ept-paragraph',
  'ept-paragraph-tile',
  'paragraph--type--' ~ paragraph.bundle|clean_class,
  'ept-paragraph--type--' ~ paragraph.bundle|clean_class,
  view_mode ? 'paragraph--view-mode--' ~ view_mode|clean_class,
  not paragraph.isPublished() ? 'paragraph--unpublished',
  'paragraph-id-' ~ paragraph.id(),
  'ept-tile-' ~ content.field_ept_settings['#object'].field_ept_settings.ept_settings.styles,
]
%}

and add styles variable at the end of the template:
{{ styles|raw }}

Any Drupal paragraph type with name ept_* can be extended with EPT Settings field and generate background and other design options.

Hello. I made changes to the file paragraph--ept-tiles--default.html - but for some reason it doesn’t work.....

You need to change this template:
modules/contrib/ept_tiles/templates/paragraph--ept-tiles-item--default.html.twig

{#
/**
 * @file
 * Default theme implementation to display a paragraph.
 *
 * Available variables:
 * - paragraph: Full paragraph entity.
 *   Only method names starting with "get", "has", or "is" and a few common
 *   methods such as "id", "label", and "bundle" are available. For example:
 *   - paragraph.getCreatedTime() will return the paragraph creation timestamp.
 *   - paragraph.id(): The paragraph ID.
 *   - paragraph.bundle(): The type of the paragraph, for example, "image" or "text".
 *   - paragraph.getOwnerId(): The user ID of the paragraph author.
 *   See Drupal\paragraphs\Entity\Paragraph for a full list of public properties
 *   and methods for the paragraph object.
 * - content: All paragraph items. Use {{ content }} to print them all,
 *   or print a subset such as {{ content.field_example }}. Use
 *   {{ content|without('field_example') }} to temporarily suppress the printing
 *   of a given child element.
 * - attributes: HTML attributes for the containing element.
 *   The attributes.class element may contain one or more of the following
 *   classes:
 *   - paragraphs: The current template type (also known as a "theming hook").
 *   - paragraphs--type-[type]: The current paragraphs type. For example, if the paragraph is an
 *     "Image" it would result in "paragraphs--type--image". Note that the machine
 *     name will often be in a short form of the human readable label.
 *   - paragraphs--view-mode--[view_mode]: The View Mode of the paragraph; for example, a
 *     preview would result in: "paragraphs--view-mode--preview", and
 *     default: "paragraphs--view-mode--default".
 * - view_mode: View mode; for example, "preview" or "full".
 * - logged_in: Flag for authenticated user status. Will be true when the
 *   current user is a logged-in member.
 * - is_admin: Flag for admin user status. Will be true when the current user
 *   is an administrator.
 *
 * @see template_preprocess_paragraph()
 *
 * @ingroup themeable
 */
#}
{%
  set classes = [
  'block',
  'ept-paragraph',
  'ept-paragraph-tiles-item',
  'paragraph--type--' ~ paragraph.bundle|clean_class,
  'ept-paragraph--type--' ~ paragraph.bundle|clean_class,
  view_mode ? 'paragraph--view-mode--' ~ view_mode|clean_class,
  not paragraph.isPublished() ? 'paragraph--unpublished',
  'paragraph-id-' ~ paragraph.id(),
]
%}
{% block paragraph %}
  <div{{ attributes.addClass(classes) }}>
    {% if paragraph.field_ept_clickable_tile.value and content.field_ept_tiles_link|render %}
    <a href="{{ content.field_ept_tiles_link.0['#url'] }}" class="ept-tile"
    {% if link_in_a_new_tab %}
      target="_blank"
    {% endif %}>
      {% endif %}
      <div class="inner">
        {% block content %}
          {% if paragraph.field_ept_clickable_tile.value %}
            {{ content|without('field_ept_tiles_link', 'field_ept_clickable_tile', 'field_ept_settings') }}
          {% else %}
            {{ content|without('field_ept_clickable_tile', 'field_ept_settings') }}
          {% endif %}
        {% endblock %}
      </div>
      {% if paragraph.field_ept_clickable_tile.value and content.field_ept_tiles_link|render %}
    </a>
    {% endif %}
  </div>
{% endblock paragraph %}
{{ styles|raw }}

There are some updates to hide EPT Settings fields:
{{ content|without('field_ept_clickable_tile', 'field_ept_settings') }}

And display generated styles from Design Options:
{{ styles|raw }}

If you installed module with composer, it's possible to patch this module, but it will be easier to override twig template in your custom theme:
paragraph--ept-tiles-item--default--custom.html.twig