Ahhh... A silent downvote, nearly four years after this was posted. So cute...
According to the comments, there is a problem of comprehension at the heart of this question, which might be the following.
Consider a real valued random variable $Z$. This is a measurable function $Z:\Omega\to\mathbb R$ between a probability space $(\Omega,\mathcal F,P)$ and the measurable space $(\mathbb R,\mathcal B(\mathbb R))$. The distribution of $Z$ (aka the law of $Z$) is definitely not the probability measure $P$, but the image of $P$ by $Z$, often denoted $P_Z$, that is, the probability measure $\mu$ on $\mathcal B(\mathbb R)$ defined by the identity $\mu(B)=P(Z^{-1}(B))$ for every $B$ in $\mathcal B(\mathbb R)$.
Thus, to declare that two real valued random variables $X$ and $Y$ are identically distributed means that $X:\Omega\to\mathbb R$ and $Y:\Phi\to\mathbb R$ for some probability spaces $(\Omega,\mathcal F,P)$ and $(\Phi,\mathcal G,Q)$, and that the probability measures $P_X$ and $Q_Y$ on $\mathcal B(\mathbb R)$ coincide. The fact that $P=Q$ or that $P\ne Q$ (both can happen) is quite unrelated.
To declare that two real valued random variables $X$ and $Y$ are such that $X:\Omega\to\mathbb R$ and $Y:\Omega\to\mathbb R$ for some common probability space $(\Omega,\mathcal F,P)$, as you seem to be intent on saying, one usually simply mentions that $X$ and $Y$ are defined on the same probability space.
Best Answer
Since you asked for two specific sorts of examples, here are some small ones. First, consider a sample space consisting of two elements $a$ and $b$, with the probability measure that gives both points probability $\frac12$. Consider the following two random variables $X$ and $Y$, both defined on this sample space. $X(a)=X(b)=Y(a)=0$ and $Y(b)=5$. Then the probability distributions of $X$ and $Y$ are very different, so they are not identically distributed.
Second, consider a probability space consisting of four elements, $a,b,c,d$, each with probability $\frac14$. Let $Z$ be the random variable on this space defined by $Z(a)=Z(b)=0$ and $Z(c)=Z(d)=5$. Then this $Z$ and the $Y$ from the previous paragraph are identically distributed, but they are defined on different sample spaces.
(If you want really small examples, both sorts of examples can be built using $1$-element sample spaces, but then there's not much probability or randomness visible.)