The main problem here is WordPress COM and ORG are NOT equal. Questions about support for the former are actually off-topic on WordPress Development.
One consequence of the non-equality of com and org is that the latter has no shortcode [instagram]
, so naturally it won’t work – the built-in shortcodes can be seen at this list. On the other hand, you are right, according to the codex page about embeds Instagram is supported. Normally it works like this:
Instagram photo below
http://instagram.com/p/bNd86MSFv6/
Instagram photo above
It is important that the embeds link is on a separate line.
If that is not working for you, then you can try using the </code> shortcode</a>: </p>
<pre><code>http://instagram.com/p/bNd86MSFv6/
BTW tested the Instagram embed on a old WP 4.0 installation, it works just fine for me.