Signature security proof in the Random Oracle model As a study case, I consider the BLS signature scheme, but the following question is relevant in the general context of security proofs in the Random Oracle model. 

Let us briefly recall BLS signature scheme: 

Let $e:G \times G \to G_t$ be a bilinear group scheme. Let $g$ be a generator of the group $G$, and let $a \in \mathbb{Z}^*_p$ be a random field element. We denote by $H$ a function that serves as a random oracle. 


Keys: secret key is $a$, public key is $(g, g^a)$.
Sign: compute $m \mapsto (m, \sigma = H(m)^a)$.
Verify: check the equality $e(g, \sigma) = e(g^a, H(m))$.       


The authors prove that this scheme is secure (or more specifically - secure against existential forgery under adaptive chosen message attack) by describing an algorithm $\mathcal{A}$ that given a forging entity $\mathcal{F}$ breaks computational Diffie-Hellman in $G$. 


  The proof assumes that the algorithm $\mathcal{A}$ emulates the random oracle, meaning that every query by $\mathcal{F}$ to $H$ is actually answered by $\mathcal{A}$. My question is, isn't this a very, very strong assumption?


If instead we assume that the random oracle is some third party that both $\mathcal{A}$ and $\mathcal{F}$ can only query but not affect (like in every real-world use case) than this proof completely breaks. 

Also, if we provide a security reduction without this strong assumption, our reduction might be far more efficient.
This was discussed by Coron in 1. 

You are actually asking why the random oracle can't just be some uncontrollable ideal random oracle. 
In fact Bellare and Rogaway when introduced their Full Domain Hash scheme (FDH) in the seminal works 2,3 used this uncontrollable random oracle to analyze the security reduction for FDH.

The thing about using reductions to prove the security of, let's say a signature scheme, is that the reduction security depends greatly on the probability to break the underlying assumption, let's say the CDH assumption. So ideally, if the reduction enables to break CDH problem with probability 1 it means the signature scheme is pretty much secure as solving the CDH problem. 

But what happens if this probability is much lower than 1?  Well, the signature scheme security becomes rather weak.

As Coron mentions in 1, allowing $\mathcal{A}$ to control the random oracle gives a tighter security bound on the FDH signature scheme than the one shown by Bellare and Rogaway.

The main thing about the tighter bound is that it does not depends on the number of queries requested by the forger, but rather only on the number of queries for signatures (which in practice is much lower than the number of hash queries such forger can do). A better bound enables one to use more efficient signature scheme.

To conclude, using a specific hash function does not say that this is an assumption of the model, it is merely a way to make the bound tighter.