I do care about the life of my friends. I'm autistic too. I love to hear their stories. Although there are some which I do not care about. Like their kids. I don't like them, I don't want to see pictures, I couldn't care less about whatever dumb, funny or smart thing they did. And I tell them. "Hey dude, nice you have a kid but it's not for me." same goes for football or formula 1. You can talk to me about it, but I'll just be thinking about fun stuff and won't hear a thing you said.
I prefer to talk to people and hear their stories when it's something I'm interested in. It generally helps to ask about those things, stimulating them to talk about something you actually like. But sometimes people just need to vent stuff and it's good to listen, even though it doesn't really interest you. But giving them a moment for it makes them feel good. But instead of acting like you like it so they will continue to talk about it, after giving them their moment for a bit, change the subject to something you like. At least that works for me. And now my friends know what I'm interested in, and likewise, so we know what we can talk about and do so neither of us gets bored.
For instance, some like to squash. I don't. Some also like to play boardgames, like me. They don't ask me to join them for squash, but they do ask me to play a boardgame. Some of my friends talk about football but they know I hate it so they talk about world domination to me instead (/jk).
If you have no common ground with someone, I don't understand how you can be friends. Everything in the relationship you have with someone would be fake. But if there is common ground, you should focus on that and be honest about what that is for you. Friends become better friends when they know the real you and accept you for who you are. If not, they are bad friends.