Despite what badly written pop-science explanations of quantum computation may tell you, a qubit is not two classical bits and $N$ qubits is not $2^N$ bits. Qubits are fundamentally different from bits, pairs of bits or anything else (except for a representation of $\alpha$ and $\beta$ in binary to precision sufficient to classically simulate the computation to whatever precision is required).
As for why people talk about $N$ qubits being equivalent to $2^N$ bits, it's hard to say. To paraphrase Tolstoy, all correct explanations are alike but all incorrect explanations are incorrect in their own particular way. Perhaps it comes from the widespread mistaken belief that quantum computers somehow try all the options in parallel; perhaps it's a mistranslation of the fact that quantum computers offer exponential speedup over some classical algorithms. Scott Aaronson has some thoughts along these lines:
Fueling the belief that countless more quantum algorithms should exist (or that are not finding them is a failure), seems to be the idea that a quantum computer could just “try every possible answer in parallel”.
But we’ve understood since the early 90s that that’s not how quantum algorithms work! You need to choreograph an interference pattern, where the unwanted paths cancel
The miracle, I’d say, is that this trick yields a speedup for any classical problems, not that it doesn’t work for more of them