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?
As a fellow NT, this hits it on the head. This can be considered rude for exactly what you say: it can come off as one-upmanship or an inability to not have the conversation be about you. Obviously, the flow of conversation, relationship, tone and topic can change just what exactly this feels like, because there are situations where it can come off as empathetic or an attempt to showing common ground. Like most things in human interaction, there are not hard and fast rules. But if OP needs hard and fast rules because of having trouble parsing things like tone and flow, I would err on the side of not sharing personal story unless specifically asked.
An assumption of needing hard and fast rules because deficient parsing of tone and flow is peak NTness.
And honestly communication does come with pretty stern (unwritten) rules, the people doing them just don't necessarily notice them, because they are very automatic and given. The NT rules have a lot of aspects like face-saving that can be entirely foreign to ND people. And on the other hand the upfrontness of ND people can be very umcomfy to NT people.
The rules are only noticed when someone does not follow them and this is where people with different communication styles get into trouble.