Let $K$ be a field which is a (transcendental) extension of $\mathbb{C}$. Let $L_1, L_2$ and $M_1, M_2$ be two field extensions of $K$ (not necessarily algebraic) such that $$L_1 \otimes_K L_2 \cong M_1 \otimes_K M_2$$ My question is: Is one of $M_1$ or $M_2$ a field extension (not necessarily finite) of $L_1$?

Not necessarily.

Let $p_1,p_2,p_3,p_4$ be primes, and let $A_1,A_2,A_3,A_4$ be extensions of $K$ of degrees $p_1,p_2,p_3,p_4$ respectively.

Let $$L_1 = A_1 \otimes_K A_2$$ $$L_2 = A_3 \otimes_K A_4$$ $$M_1 = A_1 \otimes_K A_3$$ $$M_2= A_2 \otimes_K A_4.$$

Then neither $M_1$ nor $M_2$ can be an extension of $L_1$ or $L_2$ because the degree of $M_i$ is never a multiple of the degree of $L_j$.