If you are the owner of a Trados Studio 2017 single-user license key (Freelance/Plus, Professional) and only if you upgrade this license key to a Trados Studio 2019 single-user license, you will be entitled to run both versions of software (2017 and 2019) on the same computer by activating only the Trados Studio 2019 software with the correct Studio 2019 license key. In case of a Trados Studio Freelance Plus license you can run both softwares on 2 different computers following the steps further below. 

NOTE!  This functionality is NOT available for Studio Starter and Network licenses (Professional or Workgroup).
              This functionality is NOT available if you did NOT upgrade your Trados Studio 2017 license to Trados Studio 2019

Additionally, this is possible only if your Trados Studio 2017 software is at least updated to: SR1 CU12 - build number 14.1.10012.29730 which was released on the 31st of July 2018.

If it happened that you did not install the CU12 or a later update before deactivating Trados Studio 2017 and upgrading your license to 2019 version you can manually download and install them. Check the Downloading and Installing Cumulative Updates section (scroll down to the bottom of the article where you can find an FTP link to download the latest updates) from the below article:  
https://gateway.sdl.com/apex/communityknowledge?articleName=CUs-Studio2017SR1

1. Download and install  Trados Studio 2019 - download available in your SDL account > Products&Plans > Downloads > SDLTradosStudio2019_29074.exe > hit the Download button
2. Activate it using your  Trados Studio 2019 single-user license key
3. Close Trados Studio 2019 software without deactivating the license key
4. Launch Trados Studio 2017 software

It should skip the activation phase without giving the activation window. If you check in Help tab > Product activation you should have the below information:





NOTE!  If your Trados Studio 2017 will launch in Trial mode (it will happen if you install the software for the first time on a computer) it will pick-up the licensing from Studio 2019 only after the Trial period expires. This will happen automatically and no further steps are required. The Trial version is equivalent, in terms of options and features, with a full Professional license.

As long as your  Trados Studio 2019 software is licensed your Trados Studio 2017 version will be licensed as well. 
Be aware that you won't be able to run both softwares simultaneously with the same license key.