Please do not think that the auto-suggestion mechanism is too much of any indication. It is a simple process that has no significant tie to search.
But your question is still a good one.
Both domain names will perform for search as they should. There may be some confusion for users who are looking for one or the other domain with the auto-suggestion mechanism and it will likely be that both domains will get traffic not intended for that domain. This happens all the time.
I found that a significant amount of search traffic is required to recognize a domain name as a brand resulting in placement in the auto-suggestion mechanism. Generally speaking, unless your site is absolutely on fire, 6 months is not enough time to expect this to happen. Even then, a popular domain name can pop-in and pop-out of the auto-suggestion list based upon traffic at any given point. If your site is receiving moderate traffic, do not expect it to necessarily show up in this list.
Now for the good news! If your site does increase in traffic and you do manage to listed in the auto-suggestion list, your domain name may/will likely show up before the other domain name which is longer. Now for the bad news. That may also increase your bounce rate if users are too quick to choose your domain name and not the other as intended. I cannot imagine it changing much though.
So you see, you may be better off with things the way they are. This is not something I would concern myself with too much just yet.
Now to part two of your question. Branding.
It takes quite a bit of search traffic to get recognized as a brand by Google. The only way to get this to happen is to: one, build a terrific site that people want to link to; two, compelling SERP links using good
title tags and
description meta-tags; and three, create fantastic content that people are searching for. When you are able to generate enough traffic, then Google will begin to recognize your domain name as a brand that people want.
As for site links, these will come with site branding and will only show up when a search for the brand exists. In this particular case, Google must recognize it as a strong brand and other elements must be in play including the proper mark-up/code to make it happen. Make sure you have done your homework and put into place everything that Google wants to see, then build your brand through social media, back links, satisfying search results, and high metric scores for user satisfaction. Then, and only then, will you be able to see your domain name in the auto-suggestion list and site links. Until then, it really isn't the end of the world. I would not be too hasty and jump ship to another domain if that is what you are thinking.