Unique risk neutral measure for Brownian Motion

For a standard geometric Brownian motion model of stock prices:

$$dS = a S dt + \sigma S dZ$$ we can transform the process to be under risk neutral measure: $$dS = r S dt + \sigma S d \tilde{Z}$$ and from the references I found, this risk neutral measure is "unique".

If we make a transform, say

$$dS = r S dt + \tau S d \hat{Z}$$ where $\tau$ is different from $\sigma$, this equation gives the correct price of stock. but Black-Scholes equation will fail as we have changed volatility.

However, for a discrete model, e.g. a tree model, if there are $n$ states of world, then we need $n-1$ assets plus cash to uniquely pin down risk neutral measure.

Question:The Brownian motion model in effect has infinite number of states and only one asset, then where does uniqueness of risk neutral measure come from?