Is Quantum Computer analog?
No, quantum computers are not the same as analog computers (at least in principle).

Analog computers simulate the (mathematical) problem to be solved by
building a physical system that obeys the same constraints/laws as the
mathematical problem. The answers are obtained by observing and
measuring the behavior of the physical simulation. Its accuracy is
that of the simulation (there may be parasitic effects), the accuracy
of the initial conditions, the setting of problem parameters in
particular, and the measurement on the result.

Accuracy may also be limited by the scale range of applicability of
the phenomena used for the simulation. For example, if the answer is
given by a level of water in some container, you may be limited by
capillarity effects (which can be accounted for to some extent) and by
the fact that measuring water level with more accurcy that the
diameter of a molecule may not be very meaningful.

I used to think that a major difference is that analog computing is in principle
based on the simulation of continuous laws, involving reals, while
digital computing operates exclusively on countable sets. But, in the light of current knowledge in computing theory, this
distinction is probably naive because I suspect that physics could be
formalized as well using only computable reals, of which there is only
a countable number.

Quantum computing will mainly allow you to do several digital computations in
parallel (to state it simply). It is always a finite cross product of
several computations, and hence stays in the countable realm. You may
think of it as the cross-product construction of an automaton that
simulates two or more computations of simpler automata (though it is
even less general than that from what I understand of it). These
finite cross product constructions never leave the countable realm.