Proof that finite automata is closed under intersection
A possible source of your confusion is that the transition functions of $M_1$, $M_2$ and $M$ are different functions, yet you call all of them $\delta$. If instead you use $\delta_1$ resp. $\delta_2$ for the transition function of $M_1$ resp. $M_2$, the definition of $\delta$ changes to $\delta((p,q),\sigma)=(\delta_1(p,\sigma),\delta_2(q,\sigma))$. You should be able to add the subscripts in the appropriate positions of your partial proof yourself.

Now, after noting that $\delta_1^*(p,x)\in Q_1$ and $\delta_2^*(q,x)\in Q_2$, the proof can be completed by a straightforward application of the definitions of $\delta$ and $\delta_i^*$.