I'd like to know if both sentences could be possible.
- We've gone / We've been going to the same dentist since we were children.
- You've worn / You've been wearing that coat for years.
Thank you in advance.
I would go with the present perfect progressive here (the second set of options).
The present perfect tends to be used to describe events rather than ongoing processes. There is no "internal structure" to events described with the present perfect. But the present perfect progressive shows that something has been ongoing and is possibly continuing now.
For example, take the sentence "I have tried caviar." This simply means that at some point in the past, you have eaten caviar, and that this has some relevance to the current state/conversation. For example, if someone offered you caviar, you might say:
I have tried caviar, but I didn't like it.
However, the progressive form, "I have been trying caviar" suggests that you have been regularly eating caviar, perhaps trying to acquire a taste for it, this could lead to a response like:
I've been trying caviar, but I still don't like it.
So because wearing a coat and going to a dentist are continuous actions that are done with some regularity, rather than single events, go with
We've been going to the same dentist since we were children.
You've been wearing that coat for years.