Does a new water tank need to be installed at the same height as an existing one for the floats to work properly?
With proper valve and control setup, you can set it at whatever height you like. Level control in each tank would be independent of the other tank.
There are multiple ways to do this. One would be to have the float switch on the 500 gallon tank, and a float valve (fed from the 500 gallon tank) on the 300 gallon tank, as well as a check valve on the output of the 300 gallon tank to the pressure pump before it joined the output of the 500 gallon tank to the pressure pump.
If it's just open pipes, sharing one float valve in one tank, you can't.
Of course, if the 500 gallon tank was larger diameter but the same height, that could work - you have not given specifics about those details.
If you want to elevate the 300 gallon tank, you'd move the pressure pump input to the bottom of the 500 gallon tank, so that water from the 300 would flow into the 500, with the tops of both tanks at the same level.