You need two steps:
1. Use the plugin https://moodle.org/plugins/view/mod_groupselect as Emma said. It has good control features for self grouping.
2. Set up in your assignment setting as below:
The two contributors did not answer the question here because the option they present is one where the teacher/admin must create groups to which students can join. However the question was about paired submissions - in which the teacher/admin does not create groups beforehand (otherwise the teacher/admin is forced to create a group for every two students in the class) - but that pairs of students tie each other to a single submission during the submission process to receive the same grade and feedback. Is this possible on moodle, and if so, how can it be done?
Only in groups. There is a plugin however that allows students to sign up for groups that would remove the teacher group creation part.
Here's the link to the plugin Emma referred to: https://moodle.org/plugins/mod_groupselect (Lets students create and select groups).