Numerical accuracy of superpositions in quantum computers
It appears that my question about "numerical accuracy" / error correction for qubits is nicely answered in the bible of quantum computation, M.A. Nielsen and I.L. Chuang, "Quantum Computation and Quantum Information", Oxford Univ. Press, 2010, chapter 10, in particular section 10.3.1. This book is great, I think that it is both very readable and very thorough!

The key point is the following: Yes, errors may be introduced during the calculation so that the superpositions becomes more and more "inaccurate" in small ways. Then, it may seem very strange that in many papers on error correction in quantum computation, authors often only correct for a few types of errors, in particular bit flip errors ($|0\rangle\leftrightarrow|1\rangle$) and sign flip errors ($|1\rangle\to -|1\rangle$). What about small deviations in superposition coefficients? The answer is: If you correct for the former kind of error, then the latter ones are accounted for as well! (Theorem 10.2 in the book.) The reason is that if you account for bit and sign flip errors, then you automatically account for any linear combination of these errors, which now includes small deviations in the coefficients as well. Hurray!

In general, when thinking about qubits, it is not a good idea focus on the fact that the superposition coefficients can be arbitrary real numbers. Sure, Shor's algorithm requires a fairly precise cancellation of these coefficients. But a better point of view is to think of a qubit as something much closer to a classical bit, which has only two discretes states $0$ and $1$, than to a high precision real number from the interval $[0,1]$. The point is that when we try to physically measure a qubit, we only ever get one of two discrete answers (albeit with interesting probabilities), so it's pretty much discrete.

If we compare analog and quantum computers, then it is fair to say that an analog computer tries to encode as much information as possible in high precisions real numbers, whereas a quantum computer has essentially just discrete bits with a little bit of "quantum extra". Then the key point to the computational power of quantum computers is not that a qubit contains real numbers, but that qubits combine by taking the tensor product of Hilbert spaces ($H_1 \otimes H_2$) whereas analog computers combine by just adding more real numbers, corresponding to the direct sum of vector spaces ($V_1 \oplus V_2$). The "little quantum extras" combine into something useful if we add many of them.

Also note that at the time of this writing, it is totally unclear whether quantum computers are really more powerful than classical computers. As far as I am aware of, there is no "genuine" (no oracles) problem that can be solved efficiently only with a quantum computer, i.e. where we have proof that a classical computer can't do it efficiently. Quantum computers certainly don't solve the $NP = P$ problem.