I thought that it would be interesting to collect into a big list various instances of isomorphic structures with no preferred isomorphism between them. I expect the examples to be interesting since it seems that in such situations choice of a particular isomorphism is frequently an important kind of structure.
For some reason all examples that I could come up with are related to some sort of self-duality, although there must be others not related to any duality, and I am especially curious about the latter.
So let me do this: I will ask some questions about these self-dualities. If most of the answers are about these, I will not add the big list tag. If there are many examples of some other kind, then I will.
The simplest and most ubiquitous one, you have already guessed it: isomorphism between a finite-dimensional vector space and its dual. The choice of isomorphism amounts to a non-degenerate bilinear form. While we are at that, let me ask this: at a first glance, the fact that such forms can be (at least in characteristic 0) decomposed into the sum of a symmetric and a skew-symmetric form is just the consequence of the fact that eigenvalues of an involution are $\pm1$. But initially, given just a nondegenerate form, there is no involution present, unless we have another such form. So how to explain that such a decomposition still exists? Or does it in fact not, and one has to speak about pairs of such forms??
My subsequent examples will be just generalizations of the first.
A finite abelian group and its Pontryagin dual. If it is an elementary $p$-group, this is a particular case of the above (vector spaces over prime fields). What about the general case? Is a choice of isomorphism, i.e. a nondegenerate pairing $A\otimes A\to\mathbb Q/\mathbb Z$ a structure that is actually used somewhere? I've heard about Weil pairings but know too little about them to figure out whether they are an instance of such a thing.
Conjugacy classes of a finite group and its irreducible representations. Again, does choice of a bijection between these two sets come up somewhere in mathematics?
What comes next are examples when an isomorphism need not exist.
An isomorphism between an abelian variety and its dual. Is this used somewhere?
A diffeomorphism/PL-isomorphism/homeomorphism between a manifold and its dual. Here I don't even know what I am asking. Does this make sense at all?
Here I know what I am asking: a homotopy equivalence between a finite CW-complex and its Spanier-Whitehead dual. Do such equivalences have a name?
Excuses to those who contributed extremely interesting answers and comments, but as it has been pointed out there actually exists a very similar question (with equally interesting answers). Besides, although closed, all of the answers will be accessible to everybody, right?