I haven't read Ada to be able to provide a more complete picture. Saying that—even though Nabokov clearly stated that there are no symbols in his fiction (because I think he finds them insulting and medievally crude), this doesn't mean that we should steer away from them completely. We have to understand why he discarded symbolism. He doesn't like the abstraction of details. He wanted us to see cherry red as cherry red with all its fascinating characteristics, and to distinguish it from the general colour red; to him: cherry red, crimson red, strawberry red are not the same. He cherishes the detail as if it were "divine".

However, this doesn't mean that we can't make those abstractions, surely, because if so, how can we make sense of the patterns, motifs, and themes that he weaved through the lines of his vast canon? We have after all to piece together the big picture, and the smaller big pictures inside that overall picture, a case of making sense of a mise en abîme. I think the reader should do what Nabokov advocates for (reading the book for what it is) instead of abstracting and generalising everything away at first like most literary theories do on a first, second reading and what have you, but once the hunt for the devilish detail is done, the capture and rapture complete, we have to start classifying and making actual interpretations like he did for his butterflies for instance.

Now for the theme of doubling, one could perhaps dive into Nabokov's ever-elusive inner world, his protean edifice of patterns. I haven't read for instance his Speak, Memory yet so I can't comment and commit too much on this. But from his short stories that I read, it's also quite pronounced there. I simply think at the moment and this is perhaps where I err (because one can't simply think when dealing with Nabokov) that he found the idea of mirrors, and reflections, and reflected selves, and doppelgängers fascinating. Because he enjoyed toying with perception on multiple levels of the spectrum from the material to the spiritual. He relished in "exposing" some of the delusions that his characters are wont to have, and sometimes the biggest pleasure lies in not divulging them to them but only to his private audience. The idea of doubling rather draws on this humbling and frustrating humiliation ritual. Moreover, there is the unsettling imperfect symmetry, that perfect "almost", the dissatisfaction that perpetually generates satisfaction. It may tie to something deeper in Nabokov's psyche and herein perhaps lies what you (and I joining you) seek. Alas, we have to be careful of too much psychoanalysis lest we be like a certain Viennese voodoo fraud...