Can the game be beaten on the first run, and is there any special interactions if you do so?

As was discussed in the comments, a video showing a run that beats the game quickly on the first run on a fresh file exists, with developer commentary over it mentioning some unique dialog.

Although the conversation mostly proceeds as normal, after Hades mentions that he always kept his temper, Zagreus' response is slightly different.

The usual text box, with added bold on the part that changes:

Well, let's see. You've berated me repeatedly and often. You've lied to me. Mustered your wretched forces to kill me, over and over. But, sure, I'll grant you that you've always kept most of your anger bottled up. All that's about to change, then?

While getting to Hades on the first run will cause Zagreus to have this response instead, with added bold on the changed part:

Well, let's see. You've berated me repeatedly and often. You've lied to me. Mustered a bunch of incompetent wretches to try to kill me. But, sure, I'll grant you that you've always kept most of your anger bottled up. All that's about to change, then?