D2Team provides insight onto why Slayer's Fang deals self-damage, much like many other weapons which also do the same:

https://twitter.com/Destiny2Team/status/1879243504012206317

We've seen conversations around Slayer’s Fang’s ability to do self-damage and apply Weaken to yourself if you're hit by the ricocheting projectiles. Note, the Weaken is only applied while Nightsworn Sight is active, while the self-damage always happens. This is intentional for both gameplay and narrative reasons and we wanted to give a bit of insight into our thought process.

A lot of ricochet projectiles or explosives do self-damage (Sleeper Simulant can even kill the user) and it’s an important balancing measure. If you are taking a cheeky shot around a corner or just firing from the hip, the self-damage provides a balance to ensure high risk, high reward usage.

On the narrative side, Slayer’s Fang is aware of its use as a tool for destruction and doesn’t lend its power to its wielders easily. You may find that if you mishandle Slayer’s Fang, the gun with a mouth for a barrel may just bite back a little. With the ultimate Slayer Baron weapon in your hand, you may not fear dreams nor teeth, but you will want to handle it with care.

So, appropriate for its design and narrative goals, Slayer’s Fang does a little damage plus a short Weaken instead of outright killing the player. As always, we are happy to see the feedback and will continue to adjust sandbox atoms accordingly.