The specific heat capacity of oil, and let's assume the maximum, is around 2 kJ/(kg K) . The specific heat capacity for water is around 4 kJ/(kg K), or twice that of oil. Source:
https://www.engineeringtoolbox.com/specific-heat-fluids-d_151.html When using a well insulated tank, nothing beats water as a storage system for heat since you can store more energy per volume. In addition, water will also absorb heat faster because of its thermal conductivity.
A 1000L hot water tank at 80C stores 100kWh of energy (source:
https://www.engineeringtoolbox.com/energy-storage-water-d_1463.html). You would need a tank twice the size to do the same with oil.
My place uses hydronic underfloor heating, i.e., hot water is pumped through pipes
embedded in the concrete floor (which itself acts as a thermal battery). To heat the water, you have a thermal battery which is a
3000L water tank, which is coupled to a
wood burner which can also take wood pellets if needed. Excess PV solar is also put into the tank, it's just hooked up to another heating element in the tank and switches to that when the battery is full. This is in a cold climate (63 degrees north) so hence the wood burner.