TITLE: What does "identically distributed" mean?
QUESTION [17 upvotes]: When two distributions have the same variance and shape, do we call them identically distributed, regardless of their mean (as this is usually a location parameter)?

REPLY [22 votes]: We never call two distributions identically distributed. 
We eventually do call two random variables $X,Y$ identically distributed. 
This if they have the same distribution, i.e. if $P_X=P_Y$ where $P_X$ stands for the induced probability prescribed by $A\mapsto P(\{X\in A\})$ on Borel-measurable sets $A$.
A sufficient condition for this is $F_X=F_Y$ where $F_X$ denotes the CDF of $X$.
This condition is also necessary since $F_X$ is actually a restriction of $P_X$ to specific Borel-measurable sets. This fact is less interesting, but worth mentioning as the comments on this question show.
Things like mean, variance and shape (I suspect you mean the graph of an eventual PDF) are - if they exist - completely determined by the distribution.