I don't know what's happening here. The proper way to install Telegram is to first install it on your phone, then on your computer. The version from the Solus repository simply installs a page with a QR code square, not the full Telegram application. Did you see the QR code?
Then, using Telegram on the phone, you select the menu, and Devices, and point it at the QR code that the computer displays. When it recognizes the QR code there, it installs the full Telegram application on the computer, based on the installation that's on your phone.
So, once it's on your phone, you can easily install it on as many computers as you want to, but it needs to be installed on the phone first. If there's any other way to install it on a computer without first putting it on your phone, I don't know about it. But I can guarantee that the method I just described is correct for installing it on Solus. Once I'd installed it on my phone (from the app store), I used this method to install it on four computers here.
It's really a slick system. You only need to create your profile once, when you install it on the phone, and then when you then install it on the computer(s), it's installed as a duplicate of what's on the phone - profile, all your option settings, and everything. And it only takes a couple of seconds for each computer..