Suppose that $G$ is a compact Lie group with at least two distinct irreducible 3-dimensional representations.

Can one classify those $G$ with the following two properties?

1. For any irreducible 3-dimensional representations $\pi$, the multiplicity of $\pi\otimes\pi$ at the trivial representation is 0.
2. For any two distinct irreducible 3-dimensional representations $\pi$ and $\pi'$, the multiplicity of $\pi\otimes \pi'$ at the trivial representation is 1.

EDIT: To make this question more concrete, let us assume that $G$ is semisimple of rank 2. From Jim's answer, this assumption of rank 2 is probably most relevant because we are looking at 3-dimensional representations.