There was an acknowledgement of the situation by Batman, at least. This was quite a bit later than I originally thought, in the 2001 Green Arrow: Quiver series.* When the topics of the erstwhile Arrow-Plane and Arrow-Cave come up in conversation between the two superheroes, Batman says: "Good Lord—did you ever have an original thought back then?" (As you can see, Oliver looks a bit annoyed by the question.)
The issue may also have been discussed when the characters crossed over for the first time, in the 1940s or 1950s. Unlike the first Batman-Superman crossover in Superman no. 76, early Batman-Green Arrow crossovers were not extensively reprinted.
According to Wikipedia, the similarity between the heroes in Green Arrow's earliest days was practically comical:
He retooled the concept into a superhero archer with obvious Batman influences.
These include Green Arrow's sidekick Speedy, his use of an Arrowcar and Arrow-Plane for transportation, his use of an Arrow-Cave as his headquarters, his alter ego as a wealthy playboy, the use of an Arrow-Signal to summon him, as well as a clown-like arch foe named Bull's Eye, similar to Batman's arch-foe, the Joker.
*Thanks to FuzzyBoots for locating this.