First, transporting heat as opposed to electricity is very wasteful, so you only want to transport it in very short distances.
Second, typically even small single family home requires quite large volume to store the heat effectively for many months. It isn't that complex only because in a single family home setting you already have a bunch of uncontested land available so you can use the volume that is relatively flat and not too deep.
Building this on a scale of a city would be insurmountable challenge. You would have to dig deeper than the buildings are high and any kind of works like that are difficult in urban areas.
https://en.wikipedia.org/wiki/Heat_pump
The first sentence:
"A heat pump is a device that transfers heat energy from a source of heat to what is called a thermal reservoir."
So yes, it involves energy storage.
The way this works is you store heat in the summer (warm up a lot of rock or ground underneath your house) and recover that energy in the winter by pumping a liquid through warm rock back to your house and use it as a heat source.
> While air conditioners and freezers are familiar examples of heat pumps, the term "heat pump" is more general and applies to many heating, ventilating, and air conditioning (HVAC) devices used for space heating or space cooling.
Basically what I would understand from the term heat pump in ordinary conversation would be an air conditioner intended for use in a heating dominated climate rather than a cooling dominated one, but there might be some regional differences in usage.
You can use geothermo (ground) to work around this. I'd recommend it, but the one time install costs mean it is questionable if it is cost effective.