How to show author name only if the post is in a specific category?

Yes, that is the line you need to edit, and you need to check and see if the current post is within your specified category. So you could replace that line with this code and it should do the trick:

if(in_category(90)) {
    $author = mom_option('post_meta-author') == 1 ? '<span class="author vcard" itemprop="author" itemscope itemtype="https://schema.org/Person">'.__('Posted By:', 'theme').' <span class="fn" itemprop="name"><a href="'.$author_link.'">'.get_the_author().'</a></span></span>': '';
} else {
    $author="";
}

If that doesn’t work, either you have the wrong category ID, or the post object isn’t getting set correctly at this point (i.e. you aren’t in the loop).

Also note: you don’t have to use the category ID, you can use the slug or name if you want it to be more readable code.