1 - Can you please try to enable "from all accounts / all / everything" (whatever the wording) as the "accounts to show contacts from" in the Contacts app?
This appears to do the trick, but only for one of the two phones. Strange thing though, my Exchange account is not listed as an option, just Phone, SIM, and gmail.
2 - Is Aqua Mail granted the "contacts" permission? Phone settings -> apps -> tap into Aqua Mail please -> permissions?
Yes
3 - Does the Aqua Mail Exchange account have "Contacts" listed for you, is sync enabled for it? Phone settings -> Accounts.
Yes
4 - What is the phone's Android version?
Two devices, a Samsung Galaxy Note 4 running TouchWiz with Android 6.0.1 and the other is a Nexus 6P running an AOSP variant (Pure Nexus), 7.1.1
It appears to NOT be the same result on both phones. On the AOSP variant (Nougat 7.1.1) if I choose All Contacts as you suggested, then I get my Exchange contacts even though the account is not listed. However, if I choose "custom" and select Phone, SIM, and *NOT* gmail, then I get no contacts at all. I really do not want my gmail contacts included in the list, I would really rather only have my Exchange contacts and no others.
On the Note 4, running TouchWiz (Marshmallow 6.0.1) and I choose All Contacts, I am still seeing no contacts at all. I tried clearing cache, and even data, for both Contacts and Contacts Storage. I also rebooted and still no contacts on this one.