The frequency v can be less than 1. So the energy is starting from zero, and is continuous. The black body radiation is also continuous spectrum given an arbitrary temperature T.
The fact that $\nu$ is continuous and unbounded simply tells you that, for example, a 'free' photon(boundary conditions often enforce quantization, eg standing waves) can take up any frequency whatsoever, and so, it can have any real number for energy(as there will always be a corresponding $\nu$ for it).
Now if you consider a monochromatic beam of light, which is made up of photons, then the beam can only have energies of the form-$$E=nh\nu$$, where $n$ is the number of photons in the beam. In this sense, the energy of the beam is quantised.
Finally, this holds for any kind of electromagnetic radiation. You can think of it as being made up of a discrete number of photons, and once you have assigned a frequency $\nu$ to each of them, the energy of radiation is quantized by the number of photons. There was no restriction, however, on what $\nu$ you would assign to each photon. As energy exchange by radiation occurs via these photons, this energy is called 'quantised'.
You don't have that formula quite right. It's really $$E=nh\nu$$ where $n$ is an integer and $h$ is a constant. $\nu$ can take the value of any positive real number, so the black body curve is indeed continuous. But $n$ is an integer. So the energy can take only discrete values.
It can also be explained by the fishes in a lake. The weight of fish is not quantized. There are eggs, small fishes and adult fishes in the lake. They have been growing up bit by bit. So the weight of a fish is a continuous variable. But if we look at all five kilograms of adult fishes in the lake, their total weight is of quantum distribution. If there is only one in the lake, the total weight will be 5 kilograms, if there are two in the lake, it will be 10 kilograms, if there are N in the lake, the total weight will be 5N kilograms. The quantum nature discovered by Planck is the total weight of all five kilograms fishes in the lake. This is, of course, a value with quantum distribution. This quantum property does not mean that the weight of fish is quantum value, but mean that the number of fishes is integers and the specific choice of the observer is 5 kilograms.