Is it possible to do, that the one who opens the order on admin panel in Sales > Orders becomes the one who is responsible for it? Other order managers could see that the order is taken.
There is no default option for completing your request.
However, you can have a look at this module https://amasty.com/magento-edit-lock.html
Currently it can lock editing of product pages, customers and other admin areas excluding orders. The latter can be done through customization. If you need one, please contact us.
Then how the workflow would look like. I imagine it in this way:
Result: other users can see that you locked the order but they can't enter it until it's completed.
Yes, that's right.