Why Exchange server still function improperly when primary DNS is down even though it is configured with secondary DNS?

If you're sure secondary DNS is doing its job, chance is the misbehavior is not due to DNS. A common scenario is DNS is collocated with domain controllers and Exchange server did not mange to go to the surviving domain controller.

Why Exchange server still function improperly when primary DNS is down even though it is configured with secondary DNS?

Define what you mean by "still function improperly".  What exactly is not functioning and what are your expectations?

Such as when migrating mailbox, it causes failure. ECP cannot be accessed and other malfunctioning. Also, I cannot access the project center of Project Server. I just wonder why having the secondary DNS in IP configuration won't prevent the problem when the primary DNS is down.

Hi,

Please provide your DNS configurations and ask a question in Microsoft DNS forum for your DNS issue:

https://social.technet.microsoft.com/Forums/windowsserver/en-US/home?forum=winserveripamdhcpdns

If the DNS is confirmed to be configured correctly in DNS forum, please ping your Exchange server and provide any event logs about Exchange error here for further analysis.

Regards,