You need a automation. While the formula field can calculate the name or email address of the assignee, the value of the formula field will only be text. If the value of the synced Jira field exactly matches the Airtable user name, the automation can simply copy the name from one field to the other. Note that the names must match exactly--spelling, capitalization, spacing, middle name, initials, punctuation, etc.
If the names do not match exactly, you can use a formula field to get an exact match to the Airtable user name.
You do not need a user table in Airtable unless you want one.