The simple answer is that no, you can't really get a URL that is redirected somewhere else to show up in a Google search result. I'm assuming here that the alias is setup to redirect with a 301 status response code to your full website URL.
But, in your case, that is probably a really good thing. There are a couple big reasons for that:
1) Chances are good that companyname.com has a lot of domain strength currently (links, rankings, citations, traffic, etc.). With your site living on a sub-folder, all of that weight attributed to companyname.com gets passed to your sub-folder on that domain (unless the franchise is doing something crazy with blocking your sub-folder, which I'm guessing is quite unlikely - but you can check).
As a result, if you can build a few links and setup local or social profiles also referencing your domain, you'll get your sub-folder performing quite well. Little effort, but big reward. Cool! Why fight that?
On the flip side, if you try to get everybody linking to your myshop.com your domain, you are (somewhat) starting from scratch and trying to build up your own authority and weight. There are some reasons maybe from a branding standpoint, but I'd try to benefit from what already exists. You are part of the franchise, so I'm assuming you are comfortable using their resources in other ways (processes, products, etc.), so why not use this too?
2) You'll also be more likely to get better clicks from a Google search result if the pages are indexed as companyname.com vs. myshop.com. Why? Because people looking at a search result page are often times more likely to click on a known website. It is "Oh, I know companyname.com, let's look at that" vs. "what is myshop.com, why would I go there?" You are using their brand in other ways (including to get more people to buy from you), so why not use it here too?
That isn't to say you can't use myshop.com in other marketing materials, especially when you need a short URL. But, when it comes to Google or getting links, I wouldn't worry too much about that short URL and would bias toward the longer URL.
I've helped a few other franchises in your situation and just by using the main company brand more prominently, including in the URL, we were able to get more traffic to their site and customers in the door.
Hope that helps!