I keep hearing that this is frowned upon, but I cannot help it. After I share, I circle back and explain how and why I connected the two stories to try to recenter the other person. Is this annoying? How do you want ND people to respond in that case?
Once I saw a panel discussion about some community problems. For some reason this moment really stuck with me and I think of it often.
An activist was addressing an elected politician who was also on the panel. The activist was talking about all the people they have known who have suffered due (ultimately) to policy decisions. Many people dead, injured, relationships destroyed, other kinds of harm. They were saying names of people they had lost and what circumstances had hurt them. They were crying and speaking from their heart describing the problems in the community which were clearly overwhelming.
Then the politician's turn came and they said "Oh I totally understand that, I've had it happen in my own life". Proceeded to tell a story about close friend. Friend had been driving some fancy fast car while high/drunk, got into a big crash and was charged with DUI. They were facing the possibility of substantial incarceration due to priors. However friend's parents being wealthy and powerful, they got off with no consequences. (That was literally how the politician told the story, I am not editorializing.) Ending with something like "so that was lucky."
In that moment it was a very bone-headed thing to do because the fact is that the politician simply has nothing in their life which can compare to living in a community which is marginalized and constant target of oppression as the activist had. They shouldn't have tried because it was so minimizing and made it appear the politician basically didn't hear anything the activist had said. And for whatever it is worth I bet my left pinky toe this politician is NT.
So that is an extreme example of when not to reply in kind. Some times another person's experience is so far removed from your own. Like if someone tells me about how they escaped torture and lived in a refugee camp I won't follow up with "Ya traveling is really hard" and tell a story of how I got stuck in a train station for 12 hours because I missed a connection or something.
Other times you might have a similar experience but if the other person still wants to talk, or is looking for a signal that you are interested to hear more, you can share it in an inviting way.
Example which contains potential lines of conversation, I used the example of sharing about death of a parent; nothing specific.
CW: death of parent
If someone is talking about their mother just died, they might not want to hear all the details how your mother died. But saying "I'm so sorry to hear about your mother. It sounds like a very painful time. Thank you for sharing. My mother died a few years ago. I really feel you about not realizing how much I'd miss her" and then leaving some silence lets them sort of decide where to go with it.
So if they are done talking for the moment, they might invite you to share by asking a specific or broad follow up question.
Or, if they are still needing to finish their story, they might reflect from your statement like "Yeah we had bad times but I didn't realize how difficult the holidays would be without her....." and keep going.
Or they might be completely done discussing the subject of a parent dying. Maybe they don't want to talk about either situation any further. In which case they might just reply with brief condolences, change the subject etc.
I think when someone is sharing something personal/sensitive and then the subject suddenly shifts away from their own experience, it can be a bit discombobulating. Like they are being vulnerable, opening themselves up, it's soft and tender in there. They are looking inward on themselves, but suddenly attention is shifted to another person and they have to divert their attention to digesting this other story. It can feel demanding. They also may not have gotten to the end of the story yet. Sometimes what the person really wants to share with you, comes later. but they need to tell the first part in order for you to understand the next thing. So by then going and telling something else, it detours their narrative and the conclusion is not reached.
And when you have geared yourself up to tell someone something that is really hard, but can't get there because the person basically cuts you off, it feels bad, like you are defeated. It can leave the relationship worse off than when the conversation started, rather than stronger.
My feeling is that you should introduce the subject of your own experience in a way that is initially brief, relates to what they said on an emotional level (not a factual one), and communicates that you are still interested to hear more from them. But of course context is everything.