In my original question, I asked which compact Lie groups $G$ have a certain property. Jim and Dan showed that this property is equivalent to $G$ having exactly two irreducible 3-dimensional representations over the complex numbers which are dual to each other.

There are at least three compact Lie groups that have such a property, namely $SU(3)$, $SO(4)$ and $SU(2)\times U(1)$. Let us go through these examples. There are exactly two irreducible 3-dimensional represenations of $SU(3)$ which are the standard representation and its dual. The irreducible 3-dimensional representations of $SO(4)$ descend from its universal cover ${\mathrm{Spin}}(4)=SU(2)\times SU(2)$ which has exactly two dual irreducible 3-dimensional representations $S^2(V)\otimes {\mathbf{1}}$ and $ {\mathbf{1}}\otimes S^2(V)$ (here $V$ is the standard representation of $SU(2)$ and $ {\mathbf{1}}$ is the trivial representation). Finally, the irreducible 3-dimensional representations of $SU(2)\times U(1) $ are $S^2(V)\otimes W$ and $S^2(V)\otimes W^{\vee}$ where $V$ and $W$ are the standard representations of $SU(2)$ and $U(1)$ respectively.

Okay. That was quite a lot. There are any other examples of semisimple compact Lie groups having exactly two irreducible 3-dimensional representations which are dual to each other? From Jim's answer, it is likely that such Lie groups have to be of low rank. 

The original question is given below.  

<hr />

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?
<ol>
<li> For any irreducible 3-dimensional representations $\pi$, the multiplicity of $\pi\otimes\pi$ at the trivial representation is 0. </li>
<li> For any two distinct irreducible 3-dimensional representations $\pi$ and $\pi'$, the multiplicity of $\pi\otimes \pi'$ at the trivial representation is 1. </li>
</ol>

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.