The Apollo HydroHeat system did exactly this. It used domestic hot water for space heating. They're fairly common in my area because they were sold by the gas company as an easy way to retrofit homes to gas.
(Diagram from user manual)
The system utilized a conventional-looking domestic hot water heater, except it had a higher BTU rating for the capacity. What was unique was that it had a side loop, an extra pair of water connectors on the side of the unit where a fan coil with a pump would circulate out hot water.
To guard against legionella, and for decent heating capacity, the hot water heater was supposed to be turned very hot, ensuring returned water was still hot enough to prevent bacterial growth. A tempering valve, that mixed cold water drawn into the domestic supply, was supposed to be installed, but often was not.
Apollo stopped selling the system at some point, likely due to code and safety concerns, but side loop hot water heaters are still sold and so are fan coils.
A big problem experienced with these systems is that heating boiler water is normally around 180 F. Hot water heaters can't reach these temperatures. If no tempering valve was added, the temperature has to be kept around 120 F for safety. The net result is very limited heating capacity.