The slice maps are defined in the above sceenshot.
If we take normal state $\psi$ on $N$, then the slice map $L_\psi$ defined by
$L_\psi(\sum x_i\otimes y_i)=\sum \psi(y_i)x_i$
extends to a normal conditional expectation from $M\otimes N$ to $M$.
Can we choose a special $\psi$ that make $M\otimes N$ to be $*$-isomorphic to $M$?
No. For starters, most of the time $M\otimes N$ will not be isomorphic to $N$. You could have $M=N=M_2(\mathbb C)$, and it already fails. Or if you want infinite-dimensional examples, take $N$ amenable, for instance the hypefinite II$_1$-factor, and $M$ a non-amenable II$_1$-factor, or any infinite von Neumann algebra. Then $M\otimes N$ cannot be isomorphic to $N$.
A conditional expectation is a projection, which is about the farthest you can have an isomorphism. They are not multiplicative, and they are not injective. Note that as $\psi$ is a linear functional, it will always have a significant kernel, so the slice map cannot be injective, and it can only be multiplicative if $\psi$ is, which again is something you won't usually have.