Question: Let's suppose the development team is planning the next Sprint. What does it technically mean that a User Story is added to the Sprint Backlog? Is it assigned to a developer or to the whole team? (If it's assigned to the whole development team, then how do we do this in Jira?)

Answer: 
A "story" does not necessarily correspond 1:1 with a development task. You need to determine *from* the story what needs to be done, and by whom, in order to complete it.
We have component ownership (there is a business lead and development lead for each).

We personally didn't automate the assignment of the ticket to DEV lead, but automated to assign to business lead according to the component (first component mentioned in the ticket - responsible assignee - it's set up in the Project's settings on JIRA).

Once the story is ready for development, business lead/ regular analyst re-assigns it to the proper DEV lead (in order to get rid of code ownership there is a rotation of DEV leads, so we refer to a specific table on confluence - it's updated by project manager).

It's more relevant for us, because business lead always knows the proper person, as business lead does usually collaborate with DEV lead of the component on LOE estimation, proposals to the Customer, etc.

Seems like a little overhead, but irl it's not.