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: 
The November 2020 Scrum Guide says:

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> The Sprint Backlog is composed of the Sprint Goal (why), the set of Product Backlog items selected for the Sprint (what), as well as an actionable plan for delivering the Increment (how).
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> 
> 

In this definition, the unit of work is not defined. That is, the Product Backlog Item may not be the unit of work that is Done by one or more members of the Development Team.

In my experience, most teams do one of two things. One common approach is for the Product Backlog Items to be allocated to one or more members of the Development Team. The team itself decides how this work is allocated, but the sign of a self-organizing team is frequently self-assignment and a pull-based approach. Another approach is that a team decomposes the Product Backlog Items into subtasks and those subtasks are allocated to members of the Development Team. Again, self-assignment and pulling work indicate a self-organizing team.

The exact "assignment" depends on the practices of the team and the tooling. Practices such as pair programming and mob programming have two or more people working on a single unit of work, so there isn't a single assignee. However, even in these cases, a team may opt to decide on a single person to "own" the work and they would be identified as the assignee. The tooling also matters since tools may have limitations on who is allowed to be an assignee. For example, Jira only allows one person in the assignee field, so teams may leave this blank, create team accounts, or use custom fields as a workaround.

Specifically, in Jira, you have several options:

* Add a "Team" custom field. This is generally helpful in an environment with multiple teams working on one Product Backlog but can be useful to help identify which team has taken on responsibility for the work.
* Add a "Paired" custom field. The default Assignee field can be used if there is a single owner while the new custom field can be a list of users who are pairing or mobbing on the work.
* Do nothing. If you're using the Scrum configuration with Sprints, you don't need an assignee to bring items into Sprints or move things along a Sprint Board.
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.