HttpClient cancellation doesn't kill underlying TCP call

As Damien said in the comments, HttpClient re-uses connections as much as possible, hence the reason why the connection is not closed on cancel.

When canceling a request like that, the HttpClient will just stop sending/receiving data to/from the other end. It will not send anything to inform the other end that it was cancelled. So the timeout you see of 1 minute depends on the behavior of the other end of your connection.

Also, if you want to cancel each request after 5 seconds, you can as well set the Timeout property of _httpClient to TimeSpan.FromSeconds(5). The behavior will be exactly the same (a TaskCanceledException will be thrown if the other end doesn't respond within 5 seconds).