What is the purpose of "External issue ID" field in issue form in Jira bug tracking system? Or in any other bug tracking system for that matter.
It's used when importing issues from an external system, see this answer for example
Where I've seen External ID used before is with importing tickets, migrating from systems or tools other than the current Jira instance, or relations between multiple systems when more than Jira is being used.
Just as an example, I worked in an environment that tracked their planned, prioritized development work in Jira but used Bugzilla for incoming bugs reported by both business and technical staff. The bugs would go into Bugzilla, get groomed, and future development work would be added to new Jira tickets for issues the business wanted to address. These tickets would then use the external ID field to map back to the Bugzilla issue where the defect was originally documented.