The general criterion is that no number can be found with more than one valid, distinct factorization. This might sound like I'm merely rephrasing the question, but it's actually a reframing of the question.

Plenty of numbers (infinitely many, to be precise) in $\mathbb{Z}[\sqrt{-5}]$ have more than one factorization. $6$ is just the easiest to find. To oversimplify matters, your main concern is with the "natural" primes from 2 to $p < 4|d|$ or $p \leq d$ as needed.

Now, the case of $\mathbb{Z}[\sqrt{2}]$ is actually more complicated than you might realize. Part of the complication is that $\sqrt{2}$ is a real number and so $\mathbb{Z}[\sqrt{2}]$ has infinitely many units. This sets up the trap of infinitely many factorizations that are not distinct because they involve multiplication by units, e.g., $$7 = (3 - \sqrt{2})(3 + \sqrt{2}) = (-1)(1 - 2\sqrt{2})(1 + 2\sqrt{2}) = (5 - 3\sqrt{2})(5 + 3\sqrt{2}) = \ldots$$

But $7$ really does have only one distinct factorization in $\mathbb{Z}[\sqrt{2}]$, as you can see by dividing these numbers by $1 + \sqrt{2}$, and $\mathbb{Z}[\sqrt{2}]$ really is a UFD. But the full explanation may require me to make several assumptions about what you know.

Let's look at a "simpler" domain, $\mathbb{Z}[\sqrt{10}]$, though it certainly has some of the same traps as $\mathbb{Z}[\sqrt{2}]$: $$31 = (-1)(3 - 2\sqrt{10})(3 + 2\sqrt{10}) = (11 - 3\sqrt{10})(11 + 3\sqrt{10}) = (-1)(63 - 20\sqrt{10})(63 \ldots$$

You have to look at numbers that are already composite in $\mathbb{Z}$ to begin with. And if $d = pq$, where $p$ and $q$ are distinct primes, the choice of where to look first is obvious: $$10 = 2 \times 5 = (\sqrt{10})^2.$$

Verify that $$\frac{\sqrt{10}}{2} \not\in \mathbb{Z}[\sqrt{10}], \frac{\sqrt{10}}{5} \not\in \mathbb{Z}[\sqrt{10}], \frac{2}{\sqrt{10}} \not\in \mathbb{Z}[\sqrt{10}], \frac{5}{\sqrt{10}} \not\in \mathbb{Z}[\sqrt{10}].$$ This means that $\mathbb{Z}[\sqrt{10}]$ is *not* UFD and we didn't need to compute any logarithms or sines to come to this conclusion. (You're starting to see why integral closure matters in making these determinations, right?)

Contrast $\mathbb{Z}[\sqrt{6}]$: $$6 = (2 - \sqrt{6})(2 + \sqrt{6})(3 - \sqrt{6})(3 + \sqrt{6}) = (\sqrt{6})^2$$ but $$\frac{\sqrt{6}}{2 + \sqrt{6}} = 3 - \sqrt{6}$$ and so on and so forth. This means that $6 = (\sqrt{6})^2$ is an incomplete factorization, just as, say, $81 = 9^2$ is in $\mathbb{Z}$. But this is not enough to prove that $\mathbb{Z}[\sqrt{6}]$ is or is not UFD.

As it turns out, both $\mathbb{Z}[\sqrt{2}]$ and $\mathbb{Z}[\sqrt{6}]$ are UFDs, and what is probably the simplest, most common way of proving this requires a full understanding of ideals. Adapting the proof that $\mathbb{Z}$ is a UFD to these domains can be done, but that has its own pitfalls.