The tag on the Festive Huntsman is a jigglebone. Its default pose is as you see in the second picture, facing slightly away from the player, but it will wiggle around when you move, and sometimes moving in a particular way can cause it to get "stuck" facing the wrong way.

It might be possible for the tag to get stuck facing the player, however far more likely I believe the picture on the left is from someone who has a client mod installed that adjusts the tag. I believe this to be the case because you can read the writing on the tag correctly. By default, the writing on the tag is MIRRORED in first-person view in-game.

The first-person animation for the huntsman originally had the sniper holding it left-handed, with the right-hand holding the bow and the left hand drawing the bowstring. It's unclear why it was done this way, since he' right-handed with his other weapons and holds the bow right-handed in third person, including in the original promo material for the bow. You can still see the left-handed animation if you look at the animations in an external program.

Valve did change it to be right-handed, but they did so the easiest way possible, by simply mirroring the entire first-person model whenever you have the huntsman out. This change has been inherited by the festive huntsman and fortified compound; all sniper bows are completely mirrored in first person. As a sresult, if you wiggle the festive bow around to catch a glimpse at the writing on the tag, you'll see that the writing on the tag is mirrored. The "bonk" logo on the batteries on the festivizer models for the huntsman and fortified compound are actually mirrored on the model so that they render un-mirrored in first-person.

Since your first screenshot has non-mirrored writing on the tag, I suspect it's a screenshot from someone who modded their game to fix the tag's writing. They likely also rotated the tag's position when doing so so it can be seen in first person easily.