The subject of computable model theory gives real substance to the phenomenon you describe, and in the context of countable structures at least, takes it to the next level. The question shouldn't be merely that two objects are isomorphic (or equivalent), but "there is no known" computable isomorphism"isomorphism, but rather: you should want there provably to be no computable isomorphism. This is precisely the topic of much of computable model theory.
Some good example examples:any
Dense linear orders. Any two computable presentations of the rational order (a countable endless dense linear orderorders (such as the rationals) are isomorphic by a computable isomorphism. Thus, the rational order is computably categorical.
Atomless Boolean algebras. Any two computable atmoless Boolean algebras are computably isomorphic.
Algebraically closed fields. This is a decidable theory and therefore has computable models (in any given characteristic). Ershov proved that an ACF is computablycategorical iff it has finite transcendence degree over its prime subfield. Thus, for example, any two computable presentations of the algebraic numbers are computably isomorphic.
Some bad examples:it
It gets as bad as you could possibly want. Namely, Goncharov proved that for each n<=ω, there is a computable structure with dimension n. This means that the computable presentations split into n nonempty classes of structures, such that all the structures are classically isomorphic, but computable isomorphisms exist only within the classes and never between the classes. (See S. S. Goncharov, The Problem of the Number Of Non-Self-Equivalent Constructivizations, Algebra i Logika, 19 (1980), 621-639.)
Goncharov and others have used this method to produce examples of groups, partially orders sets, unary and other algebras of any computable dimension n. See this survey paper.
The Natural numbers (N,<) have a computable presentation in which the successor function is not computable. See Shore's article.
More generally, the spectrum of a model is the collection of Turing degrees of the presentations of that Goncharov's theorem model. Knight proved that every non-trivial structure A has isomorphic copies of any higher Turing degree. See this presentation.
This last fact provides universal examples of your phenomenon, because it shows that any nontrivial structure (group, graph, partial order, etc.) will have isomorphic copies for which there is no computable isomorphism, even with oracles for one of the structures.
Thus, I take Knight's and Goncharov's theorems as a sweeping answer to your question, at least in the case of countable structures. And to summarize more generally, the fact that there are myriad provably distinct isomorphism notions in the context of computability, I believe, is one way of looking at what your question is really about.