No, that’s not it. The reason the other guy appears as Undead while you don’t appear as Archer must be because… they have unlocked their third perk (Unlife - 50 Victories), whereas you haven’t unlocked yours yet (Elf’s Friend- 50 Victories)? However, Perks don’t work in the Arena (I never had +2 Magic or +8 Life even when I selected them, for instance), so I believe this is just a display thing (like Traits showing up on troops before you launch an Arena battle, but not actually working).
What actually happens is that Hero Class color-based magic bonuses to weapons do work in the Arena. All of them. This has already been reported in this here thread and elsewhere, by yours truly and others, but the devs have not yet confirmed whether this is a technical issue or it’s working as intended.
I’m hoping they will respond eventually, and also that it is a bug, because this is completely crushing the Arena meta right now. Just pick an AOE weapon matching your class, and congrats, you take a very easy win, as long as you fire your gazillion damage to all enemies before they do the same (most often with Burning Scythe, indeed, the player pool being so imaginative and strong at resisting the temptation to abuse the flavor of the week)… No fun.
In the meantime, your Archer can abuse this too and join the current Arena meta, if you only picked a weapon with Green, since this is the bonus you get. Obviously, your “+5 magic when using Green weapons” won’t apply to the Red & Purple Burning Scythe…
And the second guy has 1 more Magic than the first one because he’s a Mythic Hero, and Heroes get +1 Magic at level 500.
I unlocked Archer and changed just to test this, in the middle of my Crescendo Arena run, since I happened to have picked Necro as my first Hero Class to level, so far, I had doubts after your post ^^. Tssk tssk.
Now let’s compare what the text tells me (whichever Class or lack thereof I happen to have) when I look at Crescendo in the Arena’s (still über impractical in 1.09, by the way) pick-your-weapon menu:
With what I actually get as a Necro (notice I too am still just a “Hero” in text, but with the subtle Khetar icon in the bottom-right corner 
you don’t need to have the 3rd perk glitched into being displayed to get this bonus):
And… what I also get as an Archer (Forest of Thorns icon, still “just a Hero”):
And it works with any weapon as long as you match the color with your class’ bonus, see?
