Honestly, this is another major issue with c....p journalism. No where does it say who the firearm is registered to. It says he gave her the gun. If he gave her the gun and it is registered to her then the background checks would have been done on her. If he gave her the gun and it is still registered in his name. Then he will likely be in trouble because a gun registered to him was involved with a homicide which makes him somewhat liable, unless he reported it as stolen in which case he is still in trouble.
USA has different gun laws in the various states. So saying American gun laws are F....d Up is also a bit open ended. Because in California or Florida it is actually very different to say Nevada or Texas. But all have basic background checks and their own nuances. Florida you can't conceal carry for example, California you can't put a 30 round mag in your Glock, Texas you can put a 100 round drum in your Glock. There are lots of differences so it is not worth labelling them all the same.
As for gifting a gun. My wife has her competency certificate. This involves background checks and letters or recommendation and a bunch of admin and both legal and practical exams and training. I can legally give her my gun to protect herself when I am out of town. She is a competent person who knows the laws and can handle the gun. I can buy her a gun and she can get it licensed in her name as well. I see no issue. Take and Oupa giving his grandchild a family heirloom gun to learn to shoot with. Or giving your child a shotgun for sport shooting. Gifting and giving guns is more common than you believe but there are legalities and admin around it.
They work similar but while it is partly for stopping power it is also for safety of people around the shooting. A hollow point mushrooms but stays 1 piece. Means that it gets a larger surface area but has significant weight/momentum. A bullet that splinters like theirs means more surface are and less weight in each part so it has less momentum and even less penetration. Yes whatever gets hit will have more energy put into it with both but neither bullet will go through and out the other side with significant energy. So whatever energy that bullet has gets transferred to the target. This does cause more damage to whatever get's hit in the form of superior cavitation in a hollo point. Or in multiple bullet channels with cavitation on each. But beyond the target it is far safer.
For the context of safety however they are both safer for similar reasons. In a place like America or in a public area they are safer for all around the incident. If the bullet mushrooms and doesn't penetrate and come out the other side then if you hit your target, anyone behind the target is safe. Remember international gun safety rule number 4- always know your target and what is beyond it. So in a public place someone behind a public shooter or shooter's target is at less risk. If you hit the target: 1 he will likely drop and threat is eradicated and 2 the bullet won't go through and into someone standing 50m behind the target. Remember you are responsible for your bullet until it comes to rest. This is big for police and working in public areas. If your shots are on target then background people are safer because even if it does go right through, it will be so slow it will not be anywhere near as dangerous.
As per American context - Drywall houses. Here if you shoot someone in the house or miss. it will likely land in a brick wall. There if you let loose in the house- you better hope it doesn't go through the perpetrator, through the wall and into a family member. If there is a shooting here in the street, you duck behind a brick wall and you are pretty safe. There you may as well not duck if it is just some cladding and drywall between you and the shooting. So a mushroom bullet or splintering bullet won't penetrate as well. So drywall then offers a little more protection and safety again for those around the shooting. This is also why a lot will not recommend an AR15 in 5.56 for self defense. Because that bullet will not stop at close range in a house. A shotgun with a slug can kill someone 2 rooms away but birdshot will not penetrate the wall. So ya, lot's of considerations to be made.