Let $$\beta_k (x) = 1 + \sum_{n=1}^\infty b_{nk}x^n = 1 + \beta'_k (x) $$ and $$B_n = \{\beta(x) \mid m_{\beta'} \le n \}$$ where $m_\beta$ is the minimal degree of $\beta$ that is, the smallest degree $k$ for which $b_k \neq 0$ in $\beta(x) = \sum b_k x^k$.
Then $B = \{\beta_k(x) \mid k \in \mathbb N_{>0}\}$ is a family of multiplicable series if $B_n$ is finite for all $n$.
My question is: Why do the series all have to start with $1$ for this definition to work?
It looks to me like one can multiply series starting with $x$ or $x^2$ etc. too because there can only be finitely many of each power because of the requirement $B_n$ finite for all $n$.
What am I missing?
You are not missing anything in particular; one could multiply a finite number of factors by a power of $x$, and/or allow factors that are just plain powers of $x$, and the result would just be multiplied by their product (also a power of $x$). In addition, one could allow multiplying a finite number of factors by a scalar, so that their leading coefficients become different from$~1$; again this would just multiply the result by the product of these scalars.
However, your point that the required finiteness is implied in the bound on $B_n$ is wrong: it only states finiteness of the number of nonzero coefficients below a fixed bound. It is not clear just how you would adapt your definition to cope with the absence of a leading term$~1$, but it seems that the formulation you would get would allow for a product like $\prod_{k=0}^\infty(x^k+x^{k+1})$, but such a product makes no sense (unless you would somewhat arbitrarily declare it to be$~0$ because there is no term that could reasonably be nonzero).
Also, while the definition could be repaired, there is little point to doing so (as the factors involved can easily be separated out), and it complicates the description. Note that $m_{\beta'}$ is the degree difference between the first and the second nonzero coefficient of$~\beta$, and that would be harder to express when there is no term$~1$. And in practice this kind of product is almost exclusively useful when all factors have a unit constant term, which is usually also the natural combinatorial value (when the coefficient counts configurations of some kind which for parameter $0$ is a single trivial one). I have never seen an example where more than this was really needed.