Eugene Kirpichov wrote:

No, a functor is a more wide notion than that, it has nothing to do
with collections.
An explanation more close to truth would be "A structure is a functor
if it provides a way to convert a structure over X to a structure over
Y, given a function X -> Y, while preserving the underlying
'structure'", where preserving structure means being compatible with
composition and identity.

As far as I'm aware, constraints like "while preserving the underlying structure" are not expressible in Haskell.

Well, they're expressible about Haskell. I.e., for functors we require:

fmap id = id
fmap (f . g) = fmap f . fmap g

The first property is how we write "preserving underlying structure", but this has a precise, well-defined meaning that we can say a given functor obeys or it does not (and if it does not, we say that it's a bad instance). But you are correct that Haskell does not allow us to require proofs of such properties.

And indeed, some people break those properties in various ways, which some consider okay if the breakage is not observable from outside a given abstraction barrier. I'm on the fence about that...