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I am having trouble understanding representations of direct products of compact Lie groups. I am aware that:

If $V$ is an irreducible finite-dimensional representation of $G\times H$, then there exist irreducible representations $V_1$ and $V_2$ of $G$ and $H$, respectively, such that $V \cong V_1 \otimes V_2$.

Let me describe an example that confuses me. Let $G=S^1$, the circle understood as a subgroup of the multiplicative group of complex numbers, and $H=\mathbb{Z}_3$, the cyclic group of order three.

Further, let $W_1$ be a complex one-dimensional representation of $H$ given by multiplication by $e^{2\pi i/3}$. Then $W := W_1 \otimes_{\mathbb{C}} W_1$ is equivalent to $W_1$, because up to equivalence, there is only one complex one-dimensional irreducible $H$-representation.

Let $V$ be a complex one-dimensional representation of $G$ given simply by multiplication by a unit complex number.

Now consider $V \otimes W$. Even though $W_1$ and $W$ are equivalent as $H$-representations, $V \otimes W_1$ and $V \otimes W$ do not seem to be equivalent as $(G \times H)$-representations. Or at least I am unable to find the equivalence.

So this is where my confusion comes from. I would have thought that the result cited above should be interpreted as follows: write down all equivalence classes of irreps of $G$ and $H$. Then tensor products of those give all equivalence classes of irreps of $G \times H$. But if $V \otimes W$ is not equivalent to $V \otimes W_1$, then what is it equivalent to?

Could someone please clarify?

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Your mistake is assuming that $W$ is equivalent to $W_1$. Identify $H=\langle h\mid h^3=1\rangle$. Then on $W_1$, $h.w=e^{2\pi i/3}w$ for all $w\in W_1$, but $H$ acts on $W$ by $$h.(w\otimes w')=(h.w)\otimes(h.w')=(e^{2\pi i/3}w)\otimes(e^{2\pi i/3}w)=e^{4\pi i/3}(w\otimes w')$$ for all $w,w'\in W_1$.

In fact, $H$ has three inequivalent irreducible representations, one for each root of the polynomial $x^3-1$ (i.e. $1$, $e^{2\pi i/3}$, and $e^{4\pi i/3}$).