Thanks for reaching out !
To activate Visual Studio Professional 2019 offline, follow these steps:

1.The key must be specifically for Visual Studio Professional 2019. If the subscription portal does not list a valid key for this edition, contact Microsoft support to verify or request the correct key. 
2. Right-click the Visual Studio shortcut and select “Run as administrator”. Go to File > Account Settings >               Unlock with a Product Key.
3. If you're using command-line installation with --productKey , ensure the key is entered as a continuous 25-character string without dashes (e.g., xxxxxxxxxxxxxxxxxxxxx).
4. Ensure you're not using a pre-release version, which does not support product key activation and requires sign-in.

Well I found two possible solutions for Visual Studio 2013, 2015, 2017, 2019 and 2022.

1. Setting new product key with StorePID.exe (MS Docs).

1. navigate to VS 2017/2019 installation directory (C:\Program Files (x86)\Microsoft Visual Studio\2017\Enterprise\Common7\IDE)
2. start cmd as admin
3. execute StorePID.exe <your_new_product_key> <MPC>
4. maybe a restart

Possible Microsoft Product Codes (MPC) are:

• 06177 Visual Studio Professional 2013
• 06191 Visual Studio Premium 2013
• 06181 Visual Studio Ultimate 2013
• 07062 Visual Studio Professional 2015
• 07060 Visual Studio Enterprise 2015
• 08860 Visual Studio Enterprise 2017
• 08862 Visual Studio Professional 2017
• 08866 Visual Studio Test Professional 2017
• 09260 Visual Studio Enterprise 2019
• 09262 Visual Studio Professional 2019
• 09660 Visual Studio Enterprise 2022
• 09662 Visual Studio Professional 2022

2. Removing product key first and set new one. (VS 2015/2017/2019)

1. open register editor
2. navigate to

• VS 2015: HKEY_CLASSES_ROOT\Licenses\4D8CFBCB-2F6A-4AD2-BABF-10E28F6F2C8F and delete it
• VS 2017: HKEY_CLASSES_ROOT\Licenses\5C505A59-E312-4B89-9508-E162F8150517 and delete it
• VS 2019: HKEY_CLASSES_ROOT\Licenses\41717607-F34E-432C-A138-A3CFD7E25CDA and delete it

3. exeute a repair installation of VS 2017/2019
4. enter your new product key under Help > Register Product

It's not the best way to solve it, but faster than deinstall and install the complete IDE.

Update (2020-01-09)

• added MPCs for Visual Studio 2013, 2015 and 2019;
• extended registry solution for VS 2019;

Update (2021-12-03)

• added MPCs for Visual Studio 2022;

You should be able to find the activation key under the registry path HKEY_CLASSES_ROOT\Licenses\4D8CFBCB-2F6A-4AD2-BABF-10E28F6F2C8F\07062 . The value is encrypted though, so you need to decrypt it with the CryptUnprotectData method in the Win32 API.

I have created a small tool that simplifies this and should allow extracting product keys for Visual Studio 2015, 2017 and 2019 (only tested for 2019 professional so far):

https://github.com/terjew/VSKeyExtractor

If you have ever logged-in to your Visual Studio subscription on the Web, you may find your key in https://my.visualstudio.com/productkeys. If you have not, you need to try to logon to the website with the credentials that you have, if you have saved them somewhere.

Otherwise, I suggest to try to put the system disk of the old computer in another computer to boot from it. Windows will not be activated on that computer, but will still work well-enough for your purpose.

For more information, see Microsoft's article
Finding and claiming product keys in Visual Studio subscriptions.