Show Other Entries in Same Category
Based on your description, this should do it:

{% if entry.style|length %}
&lt;aside&gt;
    &lt;h4 class="text-center"&gt;Related Posts&lt;/h4&gt;
    &lt;div class="columns"&gt;
        &lt;ul class="medium-block-grid-3"&gt;  
            {% set styleCategory = entry.style.first() %}
            {% set relatedEntries = craft.entries.section('news').relatedTo({
                targetElement: styleCategory,
                field: 'style'
            }).id('not '~entry.id) %}
            {% for relatedEntry in relatedEntries %}
                &lt;li class="text-center"&gt;&lt;a href="{{relatedEntry.url}}"&gt;{{relatedEntry.title}}&lt;/a&gt;&lt;/li&gt; 
            {% endfor %}
        &lt;/ul&gt;
    &lt;/div&gt;
&lt;/aside&gt;
{% endif %}


A few notes:

– I've wrapped the whole "Related posts" in a conditional testing if the entry.style categories field is not empty (if entry.style|length).

– Personal preference, but for readability's sake I find its usually a good idea to be specific/semantic with variable names (e.g. styleCategory instead of just category, relatedEntries instead of entries etc).

– The targetElement and field attributes for the relatedTo() parameter are not required – you could just do .relatedTo(styleCategory). Being specific is a minor performance win and avoids wrong results if you have multiple relations involving the same elements.

– The id('not '~entry.id) prevents the current entry from being returned