I've been through something similar, so maybe sharing my story will help.
Oh man, now I have big feelings, too! I just feel this so much!
Ooh, I know how to fix that! Maybe helping them solve their problem will make them feel better.
My favorite thing always calms me down, so maybe it'll help them too. I'll ask them to do it with me.
[all of the above are] misinterpreted as "making it all about you"
I'm not sure how to help, so I'll leave it to that person who looks like they do.
If I get involved, I'm gonna become overwhelmed myself, and that will take attention away from them, so it's best just to stay out of it.
When I'm upset or overwhelmed, I prefer to be left alone, so I'll bet they would like the same.
I'm not sure how to help, and I usually make it worse when I try but get it wrong, so it'll be better for everyone if I just do nothing.
[all of the above in this second section are] misinterpreted as cold and uncaring
Just because we don't show it the same doesn't mean we don't feel it.