>The more different the genetic material is, the less you care

This is sort of true but it misses that we don't actually have DNA sensors built into our eyes. Instead we rely on heuristics like the Westermarck effect where we will (normally) tend to not find someone we lived with as a child attractive regardless whether they're a blood relation or not.

We influence who (or what) is in our group through our behaviour, thoughts and associations. Look at the vast number of people who value their dog or cat over other human beings. It's unlikely their dog is closer to them, genetically speaking than any single human on Earth but they spend time and invest emotionally in their pet so they form a bond despite the genetic distance.

If you see a child being hurt it likely invokes a slightly stronger emotional response if the child reminds you of someone in your own life. Often this will be someone who looks like you/your family (i.e. is genetically similar to you) but it might be some other kid you've grown attached to who is not related at all.

So yes, we are driven by a calculating selfish gene mechanism but we're also burdened/gifted with a whole bunch of emotional and social instincts and rely on imperfect sensors not tricorders. It's why people can form group identities over all sorts of non-genetic characteristics (e.g. religion, nation, neighbourhood, sports team affiliation, political ideology, vi vs emacs, etc).

That's completely true because there are many aspects to what is "my group" and what isn't, but the key point is, people naturally care about their group more than they care about strangers. Thinking in terms of genetics provides a simple model that's good enough to explain a lot of phenomena. But yes, if you want to go deeper, you need to consider other factors - at first glance it seems like "culture" is the most important one.