Even better - install a water heater heat pump that uses the outgoing water as heat source. Then a) you can usefully use all drain water, not just from hot showers, and b) it doesn't matter if the outflow is at a different time than the inflow, because it's buffered by the tank in the water heater.
Or even better, get a heat pump that combines hot water and building heat. They're not cheap though.
I think you'd need a reservoir to hold the hot water before it drained for a heat pump to be effective when the inflow occurs before the outflow (as with a dishwasher). The reason that the coil systems are relatively efficient is that they are pulling cold water past hot. Using hot water to heat the warm water in a tank will be much less efficient.
I suspect that the reason why some drain heat exchangers incorporate a significant amount of brass, is to act as a heat reservoir.
You'd often be better off putting that money towards a solar hot water system. Thermomax evacuated heat tubes work almost anywhere, and aren't subject to the same service life limitations as photovoltaics and batteries.
Or even better, get a heat pump that combines hot water and building heat. They're not cheap though.