That is, if $G$ is the distribution of the sample, what does it test:
\begin{align} \mathcal H_0 : G(x) = F(x) && \mathcal H_1 : G(x) = F(x - \theta), \theta \neq 0 \end{align} where $F$ is some fixed continuous distribution with median $0$,
or
\begin{align} \mathcal H_0 : G(0) = 1/2 && \mathcal H_1 : G(0) \neq 1/2 \end{align}
Do they make a difference?
That is, if G is the distribution of the sample, what does it test:
If by "the distribution of the sample" you mean the ecdf, note that sample quantities don't belong in your hypotheses. If instead you mean "the population distribution from which the sample was drawn", then if $X$ is continuous, the sign test is testing the second hypothesis.
You can add an assumption of a location-shift alternative if you wish, but that doesn't mean that the assumption holds; the test itself is sensitive to all the alternatives included in that second set-up (e.g. in the sense that it should be consistent against alternatives where the median is not $0$).
Does the sign test only work on location families?
No, not at all. Note for example that if you instead had a scale alternative ($G(x)=F(x/\theta)$, $\theta\neq 1$) for a positive random variable $X$ and positive $\theta$, then taking logs ($Z=\log X$) would reduce it to a location-shift in the new variable, without changing the value of the test statistic -- it would work just as well for changes of scale. Indeed any strictly-increasing monotonic transformation of your random variable will produce the exact same test statistic (noting the change in the hypothesized median after transformation, such as the change from $\theta=1$ to $\log \theta = 0$ in the corresponding null). As far as the sign test is concerned, location and scale shift and the possibilities covered by the infinite array of other strictly-increasing monotonic transformations* are all essentially the same thing $-$ because quantiles are equivariant to such monotonic transformation.
Consequently it can work just as well for a wide variety of sequences of alternatives where the shape changes too; as long as the median changes, the test should be sensitive to it.
