I don't have Asperger's, but for what it's worth, I think that if you listen to what the other person has to say, and show that you have an interest, it's fine to describe a similar experience. Even better if you can relate your story back to theirs.
For example, pretend your friend is telling you about a tough time they're having with a class:
Friend: Oh my god, I was up studying all night for this exam, and I still don't feel prepared!
You: That sucks, I'm sorry. You must be exhausted. What class is it?
Friend: Professor X's Organic Chemistry class. It's so hard! And he's such a tough grader!
You: Yeah tell me about it, I had him last semester. He's a real hard ass. But you know what, I started going to the study sessions with his TA, and that really helped a lot. Plus, I think he's more lenient on people who go to those things. I guess he figures it shows initiative or something.
Friend: Hey that's a good idea, thanks!
Basically that's a simplified version of the kind of conversation I've had a million times. People like to commiserate. They want to hear about your experiences too, but they also want you to really listen to their problems. They want to feel like you care, and they want to care for you too, so they can have a connection.
Of course, it's not the same for everyone. And the amount of sharing really depends a lot on how close the relationship is. But in general, if someone tells you about their life, especially if it's something bad, you'll make a good impression if you ask a few questions to show interest and encourage them to talk more, then tell them something about yourself that relates to their situation, and find a way to tie it back to theirs.
But if they're telling you about an accomplishment or something like that, just let them have the moment. Say something like "Congratulations, I know how hard you've been working", but don't offer up much about yourself. They want to enjoy the moment and feel like you're proud of them/happy for them/jealous of them (depending on the situation and relationship).
And yeah, you're right, there are a lot of complicated rules. Different personalities, situations, cultures. It's hard enough to navigate as a neurotypical person. I don't envy your situation, but I do find your observations really interesting.