This is really a comment, not an answer. But since it is a not-so-short comment to many
answers together, it had to become an answer.

It has been observed that (first-order) definitional equivalences give categorical
isomorphisms, at least for categories of first order structures with isomorphisms as
their only morphisms. In my opinion the fact that two equivalent definitions of a
mathematical structure give the same isomorphisms but possibly different morphisms
(which maps between [complete] lattices should one consider: isotone? meet-semilattice
morphisms? join-semilattice morphisms? lattice morphisms? [complete join, or meet, or both,
morphisms?]) is a big virtue: it means that two definitions give really different points
of view on the same kind of structure (in a way, they formalize a kind of non-triviality
of the equivalence). This also happens for second-order structures (complete lattices,
uniform and topological spaces); the definitional equivalences are expressed in the natural
language of Bourbaki's "scale of sets" (or natural model of type theory) above the base
sets of the (multisorted) structure (detractors of Bourbaki and/or lowers of category
theory would instead speak of the topos "somewhat freely" generated by the (sorts for the)
base stets;
when the equivalence of definitions is completely constructive one can really take a free
topos, but depending of the principles of classical logic which are needed to prove the
equivalence of definitions, one considers the topos freely generated in more restricted
classes).

So in summary: syntactically defined equivalences induce isomorphisms between categories
of structures. As Hodges notes (for example in his book "model theory"), pratically
everything which in mathematics can "really" be considered a "construction" is
formalizable as a interpretation or at least a "word-construction" (and moreover it
is the syntactical form itself which shows what kind of morphisms more general than
isomorphisms are "preserved" by the construction. I understand that few lovers of
category theory would approve such a extreme syntactical view, but note that even
the "categories, allegories" book by Freyd and Scedrov insists on the
"Galois correspondence" between syntactical
and semantical aspects; I simply happen to prefer the syntactical side). From this
point of view, Hodges'remarks about (cases slightly more general than) adjunctions
among quasivarieties (and universal
Horn classes) induced by forgetful functors are related to the already given remark
about monadic adjunctions.

Besides, the book "abstract and concrete categories" by Adameck, Herrlich, Strecker
conains many examples of "concrete isomorphisms"; some of them shoulb be interesting
(and all of them, if I remember correctly, can be seen as syntactically defined as
above).

Incidentally, the three authors say that non reasonable concept of "concrete
equivalence" can be given; I disagree since cases exist where two categories can be
concretely reflected on full subcateories of objects "in normal form", and the
subcategories are concretely isomorphic [for example, take affine geometry of dimension
at least three: form affine spaces algebraically
defined by points,
group of translations, sfield of scalars one "normalizes"
to the particular case where translations
are a subgroup of the group of permutations of the points and scalars are a subring of
the ring of endomorphisms of the group of translations. For affine spaces geometrically
defined in Hilbert's Grundlagen style, the general case can be reflected onto the
"normal" case with the same set of points where lines and planes are sets of points and
incidence is the set-theoretic one]

It has already been observed that, in presence of choice, "isomorphic categories"
means "equivalent categories where corresponding isomorphism classes of objects have
the same cardinality". Freyd and Scedrov observe that, even in absence of choice,
the "correct" notion of equivalence is: to have isomorphic inflations. This means that
all usual examples of equivalence of categories induce examples of isomorphisms
(without the trick with arbitrary choices to consider skeletons, but instead using
canonical "inflations" of the isomorphism classes)