The Display As value in your Outlook contact record for that person has nothing to do with the sorting or grouping of emails, sent or received.
In Outlook you have the choice of sorting or grouping email by many different values, including the From and To names. In all email the sender has control over the name displayed as From. For example, in your Outlook Account Settings you set "Your Name" and "E-mail Address".
Those are the values that Outlook puts in the "From" field of an email for purpose of sorting. The "From" value is part of the email "header" once it's sent.
If the same person has different From names in different emails, that would indicate that they have changed their account settings, or, more likely, that they are sending emails from different machines (work, laptop, desktop, tablet, phone, etc.) and have different values set on some of them.
There are a lot of choices available in the "Arranged By" setting at the top of the email list display...
but I don't think any of them are going to give you what you want. You can arrange by "Email Account" which you would think would use the account the email is sent from which would improve things for you - but that refers to the Email Account on your end of the email - regardless of whether you sent it or you received it. Ideally there would be a choice for grouping by "Sending EMail Account" but no.
I've noticed this issue many times, but I'm pretty sure there is nothing you can do about it - at least not in Outlook 2007.