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: 
It's up to the team to create the sprint backlog based on the backlog items chosen by the Product Owner. Some teams do use sprint planning to allocate at least some items to individual developers, other teams adopt a "pull system", meaning items are not pre-allocated and it's up to each individual to pick up items off the backlog when they have the capacity to do so.

In Jira, if you want an issue to be worked on by more than one person then the easiest way is to have each developer create their own separate sub-tasks assigned to themselves.
As Bogdan already mentioned: "the team owns all of the stories", & the team should also owns the way to get those stories done. I mean pre-assigning to individual member or pulling when sprint progress are both fine.

One practice in my team is we always have individual owner at story level even when multiple members involve. The owner will be in charge (in front of the team) to facilitate all sub-tasks & relevant members to get the whole story done.