My text introduces multi-quibt quantum states with the example of a state that can be "factored" into two (non-entangled) substates. It then goes on to suggest that it should be obvious1 that the joint state of two (non-entangled) substates should be the tensor product of the substates: that is, for example, that given a first qubit
$$|a\rangle = \alpha_1|0\rangle+\alpha_2|1\rangle$$
and a second qubit
$$|b\rangle = \beta_1|0\rangle+\beta_2|1\rangle$$
any non-entangled joint two-qubit state of $|a\rangle$ and $|b\rangle$ will be
$$|a\rangle\otimes|b\rangle = \alpha_1 \beta_1|00\rangle+\alpha_1\beta_2|01\rangle+\alpha_2\beta_1|10\rangle+\alpha_2\beta_2|11\rangle$$ but it isn't clear to me why this should be the case.
It seems to me there is some implicit understanding or interpretation of the coefficients $\alpha_i$ and $\beta_i$ that is used to arrive at this conclusion. It's clear enough why this should be true an a classical case, where the coefficients represent (where normalized, relative) abundance, so that the result follows from simple combinatorics. But what accounts for the assertion that this is true for a quantum system, in which (at least in my text, up to this point) coefficients only have this correspondence by analogy (and a perplexing analogy at that, since they can be complex and negative)?
Should it be obvious that independent quantum states are composed by taking the tensor product, or is some additional observation or definition (e.g. of the nature of the coefficients of quantum states) required?
1: See (bottom of p. 18) "so the state of the two qubits must be the product" (emphasis added).