I was wondering if one can see entanglement entropy as a way to count microstates and if so what exactly are the microstates? Usually when one talks about entanglement entropy, one thinks of it in the context of quantum information and so it appears as a way of measuring the entanglement in some specified quantum state - when we restrict ourselves to a subsystem which is entangled with the rest of the system, entanglement entropy, in some sense, gives us how much information we lose due to this restriction. So we can see entanglement entropy as hidden information, which is a standard interpretation for entropy in general. But if we want to look at it as a way of measuring microstates, I don't understand what the microstates are... Any insight?
The entanglement entropy of a mixed state corresponds to the logarithm of the effective number of Schmidt states with appreciable weight in the mixture - you can think of these Schmidt states as the microstates. For example, a pure state has an entanglement entropy of $\ln(1) = 0$, while a maximally entangled state in an $N$-dimensional Hilbert space has entanglement entropy $\ln(N)$, so all $N$ "micro"-states in any basis contribute equally to the mixture.