Aurelia Popa 

What happens if you initially open Trados Studio 2022, then drag/drop your file in to initiate your translation? Are you able to continue working?

To address your point where you say

Aurelia Popa said:

When I want to open an sdlxliff file in Studio 2021 the file tries to open in my previous Studio 2019. 

I assume you are simply double clicking on the sdlxliff file directly within Windows and its at this point Trados Studio 2019 launches? 
To fix that, please see this article  that addresses how one can restore file associations for Trados Studio.

While I am keen to help you get 2022 working, to address the 2019 error you quoted 

Aurelia Popa said:

"Unexpected exception when initializing view part fragment matches".

Please try this article 

I hope this all helps, have a good evening

Lyds

Aurelia Popa 

What happens if you initially open Trados Studio 2022, then drag/drop your file in to initiate your translation? Are you able to continue working?

To address your point where you say

Aurelia Popa said:

When I want to open an sdlxliff file in Studio 2021 the file tries to open in my previous Studio 2019. 

I assume you are simply double clicking on the sdlxliff file directly within Windows and its at this point Trados Studio 2019 launches? 
To fix that, please see this article  that addresses how one can restore file associations for Trados Studio.

While I am keen to help you get 2022 working, to address the 2019 error you quoted 

Aurelia Popa said:

"Unexpected exception when initializing view part fragment matches".

Please try this article 

I hope this all helps, have a good evening

Lyds