In psychology what is the name of the condition where person A has an ailment and their friend person B takes on the ailments upon themself?

It's called a Factitious disorder.

I don't know what the psychological name would be but I have had this happen with one of my ex friends.  I have a few medical issues & every time I would say something about it, she had the very same thing.  It was hard to be her friend, she is a narcissist too & I think it is a kind of one upping me.  Frustrating.  I am no longer friends with this woman.

You are sort of talking about folie au deaux.