Does scoring rules really only apply to categorical outcomes? The wikipedia article on scoring rule says that

It is applicable to tasks in which predictions must assign probabilities to a set of mutually exclusive outcomes or classes. The set of possible outcomes can be either binary or categorical in nature, and the probabilities assigned to this set of outcomes must sum to one (where each individual probability is in the range of 0 to 1).

i.e. it seems like the outcomes can not be continuous. However, further down in the article it defines the expected score for a continuous target variable $Y$ as
$$\bar{S}(G) = \int_{-\infty}^\infty f_Y(y)S(G, y)dy$$
where $S$ is the scoring rule and $G$ is the "random variable generated from a forecast schema". I think this seems to say that the set of possible outcomes can be binary or categorical, but also continuous. Is this correct and in that case what would be an example of a scoring rule applied to continuous outcomes?

There's a few things going on here.
First, your first quote refers to the predicted distribution, but the second refers to the target's distribution. They are not the same and they don't have to be of the same "type". For example, you could imagine a phenomenon that does not actually have a continuous distribution but that you would predict with one, as an approximation, for tractability.
Second, the definition of a scoring rule is very general and neither restriction is needed: the predicted distribution does not have to be binary or categorical, and the target distribution does not have to be continuous, either. There are specific scoring rules that apply to certain types of distributions, but the concept is fully general.
In typical modern treatments, scoring rules are defined as mapping the prediction $F$ (which is a CDF) and the observation $Y=y$ to a single real number. The expected score is:
$$\mathbb{E}(S(F,Y))$$
where the expectation is taken over the distribution of $Y$. It is not necessary that $Y$ admit a density, as you wrote. Because it's defined in terms of the CDF, $F$ does not have to be of any specific "type", either.
A fairly general scoring rule is the continuous ranked probability score (CRPS), defined like this:
$$S(F,y) = \int_{-\infty}^{\infty} (F(x)-I[y\leq x])^2dx$$
It is a measure of distance between the predicted CDF $F$ and a "perfect" prediction that assigns probability 1 to the observed outcome ($P[Y=y]=1$). Despite the name, it can be applied to any type of target or predicted distribution.
If you restrict $Y$ to $0$ or $1$ outcomes, and $F$ to Bernoulli distributions, the CRPS reduces to the Brier score. The fact that the Wikipedia article is worded in this way is possibly a consequence of the fact that measures like the Brier score predate more general ones like the CRPS and are more widely known in general.
