Under what condition is P/poly equal to the class of languages having Turing machines running in polynomial length with polynomial advice?
Since $P/Poly$ talks about circuit families for languages (different circuit $C_n$ for length $n$ inputs), it is natural to talk about an advice which is a function of the input's length alone.
Different advice for each string will make the class too big. For any $L\subseteq \Sigma^*$, your advice for some $x\in\Sigma ^*$ can be 1 if $x\in L$ and 0 otherwise.