The short answer is "because you are considering $\mathbb{C}$ and $\mathbb{R}[\epsilon]$ them with different structure", which is an artificial choice.
Maybe to illustrate the point : If I want to work with $\mathbb{Q}[\sqrt{2}]$ would you consider that I can tell $\sqrt{2}$ and $-\sqrt{2}$ apart ? I'm not sure you can come up with an an argument for one answer over the other everybody would agree on.
Whether you can distinguish or not these depends on the type of structure you include. As you pointed out, "algebraically" - that is only using the ring structure - you can't distinguish $\epsilon$ and $-\epsilon$, to distinguish them you added new structure, for example the lexicographical order on $\mathbb{R}[\epsilon]$, or the way it acts differentiable functions can be evaluated on dual numbers.
Similarly, If you only consider $\mathbb{C}$ as a field, then you can't distinguish between $i$ and $-i$ because there is a field automorphism exchanging them, but you could add structure on $\mathbb{C}$ that would allow to distinguish them. For example, one could completely arbitrarily consider $\mathbb{C}$ as a field with a marked element "i", or put a "lexicographical order" on it, so that $i> -i$; that would make $i$ and $-i$ distinguished. Of course this structure is not very interesting, so you don't want to do this, and most people will not accept this as an answer.
But what about the structure of "being oriented" as a 2-dimensional real vector space (in addition to its field structure) ? That's a structure on the complex numbers that I would consider relevant, especially if you are using complex numbers to do geometry, and that allows to distinguish between $i$ and $-i$.
And on the contrary, if you consider $\mathbb{C}$ with less structure, for example as just a real vector space, then you can't tell $i$ and $-i$ apart but, you can't tell $i$ and $2i$ or $2i+3$ apart either.
For the question of $\mathbb{Q}[\sqrt{2}]$ the answer could depend on whether you are considering it as a field or as an ordered field.
So at the end of the day, the only question is whether you would consider that a certain structure that could be used to distinguish two (or more) elements is relevant or not. And that's a purely "sociological" question that has nothing to do with the mathematics, only with what we chose to consider relevant or not.
