The question is a bit confusing because it seems to be built on at least 5 misunderstandings.
Allopatry and gene flow
[..] allopatric speciation seen in the Galapagos finches. Adaptive radiation occurred because the finches flew from island to island, so there is gene flow [..]
By definition, an allopatric speciation is a case of speciation where there is no gene flow. If there is gene flow, then you may still end up with a reduction of gene flow and a speciation event but that would not be called allopatric speciation.
The first event leading to isolation (that is to an allopatric scenario) might be caused by birds flying over to another island but this does not mean that the divergence between populations occurred with gene flow.
[..] the different environments on the different islands provide different selective pressures that causes relative fitness of the same species of birds to change.
It makes little sense to talk about relative fitness of a species. The relative fitness of an individual is the absolute fitness of this individual divided by the absolute fitness of a reference individual (generally a theoretically super fit individual) in the population.
Did you want to talk about competition between species? I don't think so as I don't know how this concept would fit in the rest of the question.
Assumption of similar ecology
[..] wouldn't it be safe to assume that these islands generally have similar types of plants, similar types of insects, similar climate, similar predators.
No, it really is not a good assumption. In some extreme cases, climate can actually radically differ between two locations at less than 20 meters apart! The same is true for other variable making up the ecological conditions.
Species dependent migration abilities
If gene flow between populations of finches can occur across islands, why can't it happen for the other biotic factors "contributing" to the islands' characteristics?
Because birds have wings! It is easier for a bird to fly to the next island than it is for a mouse.
Implicit assumption that selection is needed for speciation
So how can such similar conditions/environments provide selection pressure that is different enough to result in speciation?
Speciation can happen in absence of divergence selection (esp. in allopatry). Genetic drift alone yield to population divergence too and can lead to reproductive isolation.
To my knowledge, the relative role of selection vs drift in causing speciation is not well understood though.