In which sense we have unicity of solution in stochastic differential equation?

Question

Let $$dX_t=f(X_t)dt+g(X_t)dB_t,$$ and SDE with $f$ and $g$ nice enough to have existence and unicity of the solution. I'm not sure what mean unicity here. For example, consider the equation $dX_t=dB_t\quad \text{and}\quad X_0=0.\tag{E}$ Does it mean that $X_t$ is a standard brownian motion, or do we really have $\mathbb P(\forall s, X_s=B_s)=1$ ? (or maybe rather $\forall s, \mathbb P(X_s=B_s)=1$). Or does it mean that $(X_t)$ and $(B_t)$ has same finite dimensional distribution ? For example, $(tB_{1/t})_t$ is also a solution of $(E)$ ?