How do I change the Chat owner, on an account that is integrated with Zendesk Support?
On integrated Chat accounts, agent roles in Support are reflected within Chat. This means that the Chat owner is the same as the Zendesk Support owner. This rule extends to administrators as well, only Support administrators will show up as Chat administrators. Unfortunately, there is no way to adjust the owner or admin roles within Chat for integrated accounts, as these rules are hard-coded and cannot be adjusted or modified within the Chat dashboard.
For more on updating the account owner, see this article: Changing the account owner.
For more on updating agent roles, see this article: Setting roles and access in the Zendesk Admin Center.