This is not even true for finite groups, in this generality, and not even in characteristic $0$. Consider, for example, the group $Q_8 \times C_3$, where $Q_8$ is the quaternion group and $C_3$ is cyclic of order $3$, and consider $\mathbb{Q}$-representations of this direct product. The standard representation $\rho$ of $Q_8$ is not realisable over $\mathbb{Q}$, only $\rho\oplus \rho$ is. $C_3$ has an irreducible $\mathbb{Q}$-representation $\chi$, given by the sum of the two non-trivial irreducible complex characters of $C_3$. Now, $\rho\otimes \chi$ is realisable over $\mathbb{Q}$ and defines a simple $\mathbb{Q}[G]$-module, but it is not of the form $V\otimes W$ for any $\mathbb{Q}[Q_8]$-module $V$ and $\mathbb{Q}[C_3]$-module $W$.

If you wanted to restrict to representations over $\mathbb{C}$, then the statement will be true also for profinite groups, because any continuous complex representation of a profinite group will factor through a finite quotient.

This is not even true for finite groups, in this generality, and not even in characteristic $0$. Consider, for example, the group $Q_8 \times C_3$, where $Q_8$ is the quaternion group and $C_3$ is cyclic of order $3$, and consider $\mathbb{Q}$-representations of this direct product. The standard representation $\rho$ of $Q_8$ is not realisable over $\mathbb{Q}$, only $\rho\oplus \rho$ is. $C_3$ has an irreducible $\mathbb{Q}$-representation $\chi$, given by the sum of the two non-trivial irreducible complex characters of $C_3$. Now, $\rho\otimes \chi$ is realisable over $\mathbb{Q}$ and defines a simple $\mathbb{Q}[G]$-module, but it is not of the form $V\otimes W$ for any $\mathbb{Q}[Q_8]$-module $V$ and $\mathbb{Q}[C_3]$-module $W$.

If you wanted to restrict to finite dimensional representations over $\mathbb{C}$, then the statement will be true also for profinite groups, because any continuous complex finite dimensional representation of a profinite group will factor through a finite quotient.