When we are using Random Oracle Model
Let us first step back and say more precisely (mathematically) what "the Random Oracle Model (ROM)" is, because it is fundamentally different from an "assumption" as they are thought of in cryptography.

One way to formalize the ROM is to see it as alternative model of computation.  In particular, algorithms in this model of computation have a extra distinguished state (or subroutine, or circuit gate, whatever), that computes some function $H:\{0,1\}^\ast\to\{0,1\}^n$, where n is the security parameter.  An algorithm A is executed in this model by picking $H$ at random from all possible functions, and then running $A$ with its $H$ subroutine set to compute that function.

When we "analyze a protocol in the ROM", we mean first define the protocol via algorithms that have access to $H$, and then we prove a theorem about the protocol when it is attacked by adversaries that also have access to $H$.  The theorem will say that something happens with high probability, where the probability is over the choice of $H$, amongst other things.

This model of computation is not realistic, however, because we do not all have access to a truly random function.  So instead we take our algorithms (which were defined in the ROM) and set $H$ to something efficiently computable, like SHA-256.  Of course, the theorem does not actually cover such usage, but this approach is remarkably effective for avoiding wide classes of attacks against protocols.  We just hope ("assume") that things will work out for the protocols under consideration, but in general this assumption is false[1].  

A cryptographic assumption, on the other hand, say something about the limitations of the "standard model of computation", i.e. Turing machines, circuit families, etc.  A distinguishing feature here is that assumptions might true, but the ROM will never "true".  This holds even for interactive assumptions that involve weird oracles - even a random oracle.  

Things brings us to the point of confusion:  When you're sitting down to design a protocol (algorithm) and prove it secure in the standard model under an assumption, you can't assume that your algorithm has access to any oracles, because it needs to be an algorithm in the usual sense.  Even if your assumption has a random oracles represented somewhere in it, you still need to give a well-defined algorithm or protocol.

[1] See "The random oracle methodology, revisited" by Ran Canetti, Oded Goldreich and Shai Halevi.