No, you shouldn't connect your URL shortening domain to Google Search Console / Webmaster Tools.
You are using your domain entirely as a URL shortener, which means the main domain, and all URLs stemming from it, are 301 Redirected to other destinations.
This literally means that you are already telling Google that your pages hold no standalone value and only the destination pages do.
In that way, you cannot have a URL from this domain rank in Google.
That does not mean that these links hold no SEO value. For example, if you use your branded short link in one article on your site, and it points at mybrand.link/other-blog-post which then redirects to mybrand.com/other-blog-post, it will operate almost the exact same as if you did not have the intermediate step. It will pass nearly the exact same amount of "link juice."
The only thing not happening is the "page" mybrand.link/other-blog-post will not rank in Google. This is because it is essentially a blank page with only one thing "written" on it, which says:
"Google and users, please be forwarded to this article."
This article is about:
- Google Search Console / Webmaster Tools and Rebrandly
- Link SEO
- 301 Redirects