Many market models assume it because it makes the maths tractable: option pricing, hidden Markov regime models, and most reinforcement-learning setups all lean on it. It is an assumption of convenience, not an observed fact.
Markets clearly violate it in places. Volatility depends on the last several weeks, not just today; the memory of a recent crash changes behaviour for months; and positioning built up over time affects how the next shock propagates.
When you widen the definition of state to include those things, realised volatility, positioning, time since the last shock, the property can approximately hold again. Model design is largely the choice of what to put in the state.
Related: martingale-process, market-regime, feature-engineering, random-walk