Aww, I believe there's nothing wrong with you. You're just very empathetic and you have a kind and caring soul. Those are beautiful qualities and never make yourself feel bad for those traits.
You are not alone though! I get that way myself whenever I play Read Dead Redemption 2 Online! I refuse to kill coyotes because they sound and act too much like a dog. Whenever I hear them whimper, I feel sad so I avoid them always. My sister on the other hand has nooo problem with ridding them lol
I believe certain aspects of ourselves are discovered through situations we associate with on the daily and it (sometimes unbeknownst to them) helps people identify closely how they themselves perceive the world and their connection to it. Sometimes, others feel things very deeply and sometimes, others may not, and that's perfectly fine! Nothing wrong about that. Everyone is different after all!
Sometimes random scenarios just click stronger and you might tend to empathetically connect to the situation much more intense than others depending on how sensitive one is. It just shows your emotional intelligence is very strong and active. It can make you feel odd or even isolated because most don't tend to talk or show what may or may not affect them, making it hard to identify with what is "normal" but you'd be surprised to know how many people feel the same as you do.
If shiny hunting however does make you sad, perhaps staying away from such is a good idea so you don't have to feel sad whenever you play. You always take care of you first as you know your limits and boundaries. Know that you always have full control over you and what you can do to help protect your energy to keep it on the up and up if you can. Gaming should be fun after all! Also, yes, your pokemon will not attack a shiny as of a recent update iirc! Very much a needed thing!
There are other methods to shiny hunting too if you don't care for doing it that way! Breeding via Matsuda Method, random encounters or even looking for some in trades are a few ways to do it! I know that the knockout picnic method is quickest but with enough patience you'll get your shiny if determined enough! Just take your time with it and I know you'll get your shiny!