How do you process the death of someone?

I'm the exact same as you, I cry at sad stories online, movies, and even music lol, but I go numb when something really sad actually happens IRL, like when my dog was put down or when my dad died.

I've always been this way, and I'm 33 years old now. I worried about it when my dad passed away, I wanted to feel something but couldn't for quite a long time. I was in distress, I was in a dark place mentally, but I couldn't cry or even feel sad about him. I finally felt grief when one day I saw a man walking with his daughter sitting on his shoulders, like my dad did with me when I was a child. After that the feelings of grief would come and go.

At this point in life, I have come to the conclusion that it's not really that strange, or even unhealthy. I believe the most important thing to realize for people who process grief this way, is to recognize that they still do grieve, even if it's not coming out as sadness. It still affects the person and causes stress, so it's important to give yourself time and space to process what's happened, regardless if you cry or not.