Even though it is perhaps not surprising for the applications to repns of reductive Lie groups or reductive adele groups, and of reductive p-adic groups, yes, irreducible unitaries of products $G\times H$ are (completed) tensor products of irreducible unitaries of the factors. However, I think this is not "trivially" true, because it depends on showing that these groups are "type I", meaning that "factor repns" are actually isotypic. Many naturally-occurring groups fail to have this property. This property for p-adic reducitve groups was completely proven (in the supercuspidal case) only as late as 1974, by J. Bernstein. Yes, Harish-Chandra proved type-I-ness for reductive Lie several years earlier.

But, then, granting that, the $L^2$ version of your question is affirmative.

Probably the more general assertion (about spaces of moderate-growth functions) has an affirmative answer, but I would not know how to prove it quickly.

The abstract question about decomposition of irreducible repns of products on larger classes of TVSs is essentially open, I think, even for reductive groups, unless something has happened in the last decade or two.

Edit: for examples like theta kernels, it depends partly on how one chooses to define the things. Even in simpler situations, outside of $L^2$ there is a loss of "semi-simplicity", for example, residues of Eisenstein series are quotients with typically uncomplemented kernels. If the Segal-Shale-Weil/oscillator repn is taken to be the *unitary* one, then since the groups are type I there is a direct integral decomposition into isotypic components (factor repns), on general principles. There is no a-priori guarantee about multiplicites... although many results are known (Howe-conjecture things, first-occurrence stuff due to Kudla-Rallis and others) about the structure.

Edit-edit: ... which reminds me of a hazard: for example, with real-anisotropic orthogonal groups larger than the symplectic groups in pairings, the trivial repn of the orthogonal group maps (Siegel-Weil) to a copy of a holomorphic discrete series containing Siegel-type Eisenstein series. The point is that the repn is unitary, but the Eisenstein series is not in automorphic $L^2$. Of course there is no paradox, but there is some risk of saying irrelevant/silly things. A similar minor hazard is already present for non-compact arithmetic quotients, since Eisenstein series enter the spectral decomposition "continuously", are not in $L^2$ individually, but of necessity generate unitary repns "abstractly", as would any "generalized eigenfunction" entering a decomposition of a Hilbert space.

Another edit: About Type I... Certainly one direction, that a tensor product of irreducible unitaries of $G,H$ is irreducible unitary, is not difficult. It is the other direction, that an irreducible unitary of $G\times H$ necessarily factors as (completion of) $\pi_1\otimes \pi_2$, that requires something (I'm pretty sure!) In looking at the natural argument to prove such a factorization, at some point one finds that the big repn restricted to $G$ has endomorphism algebra with center consisting only of scalars. *If* we can conclude that the repn is isotypic (=sum or integral of a single irreducible), then the factorization will succeed. Otherwise, the argument stops, in any case. Alain Robert's book on repns of locally compact groups mentions some not-type-I groups. I do not know enough about them to make a counter-example to a claim that this hypothesis is not necessary. Indeed, conceivably the failure of the proof mechanism does not deny the conclusion...?