I have never heard the gauss markov assumptions (the classical assumptions) raised in the context of ecm (which I freely admit I am new to). In as much as it is regression, and I assume it is although its rarely called that, then they should apply.
ECM exists primarily to deal with non-stationarity which is not an issue raised by classical assumptions. Regression did not deal with time series, or did not consider its issues, when the classical assumptions were built. Time series commonly breaks a key assumption of regression, independence since there is autocorrelation in the error term. But ecm deals with a much more serious issue spurious regression. Again I don't think the classical models were built to deal with trends in data - they were largely cross sectional in nature.
I am learning ardl cointegration/ecm something that is pretty scary. Are you doing ECM in the context of VECM?