Let $X$ and $Y$ be two $k$-schemes ($k$ is a field ).Suppose that $\Gamma(Y,\mathcal{O}_Y) =k$. Is the true that $$\Gamma(X \times_{spec(k)}Y, \mathcal{O}_{X \times_{spec(k)}Y}) = \Gamma(X,\mathcal{O}_X) \otimes_{k} \Gamma(Y,\mathcal{O}_Y) $$
PS: The categorical argument claiming that global section functor preserves colimits is incorrect as explained in one of the answers. In general the above statement is false.
On
The argument you make is not correct.
As you say, the prime spectrum functor $\mathrm{Spec} : \mathbf{CRing}^\mathrm{op} \to \mathbf{Sch}$ and the global section functor $\mathscr{O} : \mathbf{Sch}^\mathrm{op} \to \mathbf{CRing}$ form a contravariant adjunction on the right:
$$ \hom_{\mathbf{CRing}}(R, \mathscr{O}(X)) \cong \hom_{\mathbf{Sch}}(X, \mathrm{Spec}(R) ) $$
which means that both $\mathrm{Spec}$ and $\mathscr{O}$ send colimits to limits. The formal argument is
$$ \hom_{\mathbf{CRing}}(R, \mathscr{O}(\operatorname{colim} X_j)) \cong \hom_{\mathbf{Sch}}(\operatorname{colim} X_j, \mathrm{Spec}(R) ) \\ \cong \lim \hom_{\mathbf{Sch}}(X_j, \mathrm{Spec}(R) ) \\ \cong \lim \hom_{\mathbf{CRing}}(R, \mathscr{O}(X_j) ) \\ \cong \hom_{\mathbf{CRing}}(R, \lim \mathscr{O}(X_j)) $$
and thus $$ \mathscr{O}(\operatorname{colim} X_j) \cong \lim \mathscr{O}(X_j) $$
This shows that you have the direction wrong; the assertion you hoped to make was
$$ \mathscr{O}(\operatorname{lim} X_j) \overset{?}{\cong} \operatorname{colim} \mathscr{O}(X_j) $$
However, since $\mathscr{O} \circ \mathrm{Spec}$ is naturally isomorphic to the identity, one thing that does follow for formal reasons is that
$$ \mathscr{O}(\lim \mathrm{Spec}(R_j)) \cong \mathscr{O}(\mathrm{Spec}(\operatorname{colim} R_j)) \cong \operatorname{colim} R_j $$
I.e. the assertion you hoped to make is true for purely formal reasons in the special case that the schemes are affine.
This is true in great generality, but it is not a formal categorical thing. That the map is an isomorphism is a consequence of the Künneth formula when $X$ and $Y$ are quasi-compact and separated (or just quasi-separated). The Stacks Project tag is https://stacks.math.columbia.edu/tag/0BEC.
When $Y=\mathrm{Spec}(B)$ is affine and $X$ is quasi-compact and separated, the result is a special case of the compatibility of cohomology of quasi-coherent sheaves with flat base change, and can be proved by tensoring the first few terms of the Čech complex of $\mathscr{O}_X$ relative to a finite affine open covering of $X$ with $B$. All that matters for this argument is that the base is affine and that $B$ is flat over the base.
I would think that the argument given in the Stacks Project becomes simpler if all one wants is the degree zero isomorphism for structure sheaves, but I have not read it carefully enough to say anything precise.