Display list of only specific tags on product

First: you’re using = which assigns a value; use == to compare the values (or === to compare strictly, ie, match both the value and the type).

Second: Your code snippet shows that you’ve mismatched the "s in the array, which should probably result in a PHP fatal error. I’m assuming that’s an error in the posted code but not in your production code.

Finally: your comparison will only work if $product_tag->slug matches the entire array shown. If you only want to check if $product_tag->slug matches one of the items in the array, you’ll need to use in_array().

eg:

if ( in_array( $product_tag->slug, ["TAG1", "TAG2", "TAG3"] ) ) {
    $img_url = get_option( 'z_taxonomy_image' . $product_tag->term_id );

    $this->format_setting( $product_tag->name, $img_url, $product_tag->term_id );
}