If you are the owner of a Trados Studio 2019 single-user license key (Freelance/Plus, Professional) and only if you upgrade this license key to a Trados Studio 2021 single-user license, are you entitled to run both versions of software (2019 and 2021) on the same computer.

Please make sure you are only activating your Trados Studio 2021 software with the correct Studio 2021 license key. In case of a Trados Studio Freelance Plus license, you can run both software programs on two different computers following the steps from the Resolution area below. 

NOTE!  This functionality is NOT available for Studio Starter or Network licenses (Professional or Workgroup).
             This functionality is NOT available if you did NOT upgrade your Trados Studio 2019 license to Trados Studio 2021

Additionally, this is possible only if your Trados Studio 2019 software is at least updated to: SR2  CU7 -Build 15.2.7.2849, released on 7 July 2020.

If it happened that you did not install the SR2 CU7 or a later update before deactivating Trados Studio 2019 and upgrading your license to 2021 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-Studio2019SR2

.

1. Download and install  Trados Studio 2021 - download available in your RWS account > Products&Plans > Downloads >SDLTradosStudio2021_16.0.0.2838.exe > hit the Download button 2. Activate it using your Trados Studio 2021 single-user license key 3. Close Trados Studio 2021 software without deactivating the license key 4. Launch Trados Studio 2019 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 2019 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 2021 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 2021 software is licensed, your Trados Studio 2019 version will be licensed as well. 
Be aware that you won't be able to run both software programs simultaneously with the same license key.