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Your URL shortening domain does not require connection to Google Search Console / Webmaster Tools.

Using your URL completely for shortening means that main domain, and all URLs related to it, are 301 Redirected to other destinations. This literally means that you have already “told” Google that your shortened links have no standalone value, but only the destination pages do.

In that way, you cannot have a short URL ranked in Google. Notwithstanding your original links certainly have SEO value. In other words, from Google ranking point of view, if you use your branded short link in some article, it will operate as if you did not have the intermediate step. It will pass nearly accurate the same amount of "link juice".