I'm seeing that same thing on my new galaxy note 3. Here's what found:
1) The password storage type on the server was set to SHA1 which the android client does not support. (I was having the same issue in the eMClient desktop mail application showing the same data for the two different accounts). This is probably not causing you an issue.
2) I was using the same password for both accounts. (I changed on of them). As you will see in minute, this is why both accounts showed the same data.
Once I changed the pasword in one account and disabled SHA1 password storage, the eMClient mail application does now shows the correct mail information for each account. However the android client is not longer able to login successfully on the second account. Looks at logs on the mail server I found that the android mail client is sending the WRONG username for the second account. It's sending the username of the first account, but the password for the second account! So if your passwords are the same, then you get the same data in both.
This is a HUGE bug in the android mail client. Obviously multiple accounts are not part of their QA test matrix. It's not really enterprise ready apparently.