If we have two anticommutative graded rings $M=\bigoplus_{k\geq 0} M_k$, $N=\bigoplus_{l\geq 0} N_l$ (anticommutative meaning $ab=(-1)^{deg(a)deg(b)}ba$ for $M$ and the same for $N$) such that each $M_k$,$N_l$ is an $R$-module, $R$ commutative with unit, so that $M$ and $N$ are $R$-algebra, we can take the tensor product $M\bigotimes_R N = \bigoplus_{n\geq 0}\bigoplus_{k+l=n} M_k\bigotimes_R N_l$ and define:
$$ (a\otimes b)(c\otimes d) = (-1)^{deg(b)deg(c)} (ac\otimes bd) $$
So that $M\bigotimes_R N$ becomes an anticommutative graded $R$-algebra with the $n$-th degree being elements $a\otimes b$, $a\in M_k$, $b\in M_l$ such that $k+l=n$. However, it is also possible to define:
$$(a\otimes b)(c\otimes d) = (-1)^{deg(a)deg(d)} (ac\otimes bd) $$
And this also defines an anticommutative graded $R$-algebra with the same properties as above. And the multiplication is not the same because one multiplication is $(-1)^{mn-deg(b)m-deg(d)n}$ of the other (where $deg(a)+deg(b)=n$ and $deg(c)+deg(d)=m$ ) so the elements change of sign depending of the degree of the factors and tensor product.
My question is, does this two multiplication define two different structures as above? Or somehow are those two different structures isomorphic to eachother?
Your second structure is isomorphic to $N \otimes M$. Let's call $M \odot N$ your second tensor product algebra structure. The map is simply $$\tau : M \odot N \to N \otimes M $$ $$ m \odot n \to n \otimes m$$ The map is an algebra map : $\tau ((a\odot b)\cdot (c \odot d)) = (-1)^{|a||d|}\tau( ac \odot bd) = (-1)^{|a||d|} (bd \otimes ac) = (b \otimes a) \cdot (d \otimes a) = \tau(a \odot b) \cdot \tau(d \odot c)$.
Moreover, you can check $M \otimes N$ and $N \otimes M$ are in fact isomorphic by the map $$\phi : M \otimes N \to N \otimes M$$ $$ m \otimes n \to (-1)^{mn} n \otimes m$$
Check : $\phi(a \otimes b ) \cdot \phi(c \otimes d) = (-1)^{|a||b|+|c||d|+|a||d|} bd \otimes ac = (-1)^{|b||c|+(|a|+|c|)(|b|+|d|)} (bd \otimes ac) = \phi((-1)^{bc} (ac \otimes bd)) = \phi( (a\otimes b) \cdot (c \otimes d)).$
This makes the category of commutative graded algebras (CGA) a symmetric monoidal category.