Within a board, users without archive issue project permission can still see the option to archive issues

The ability to archive issues is restricted to those with the proper permissions. The design language of Jira will generally remove options or sections of the product that a user does not have the ability to access.

In this case, users without global/project admin or archive issue permissions are still able to see the archive issue option when right-clicking an issue within a board:

While they are not able to archive the issue, users are prompted to elevate their permissions:

The option to archive is removed from the issue detail view for these users.