Embedding Instagram via shortcode doesn’t work as advertised in official docs

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.