$p(x\mid C)$ is defined as the probability density of a point $x$ given that it belongs to a class $C.$ But what of $p(\mathbf{x}\mid C)$ where $\mathbf{x}$ is a vector?
I'm finding hard to understand this concept as a vector is made up of elements (i.e. $\mathbf{x}=(x_1,\dots,x_n)$.
I presume your question is in the context of the classification/discriminant analysis where you want to classify observations into classes. The concept from a single value to the vector value is nothing but just an extension to a higher dimension.
In the single-valued scenario, think as if you had a single feature (say, only weight of the individuals) available with you for performing the classification (classifying individuals among certain classes) - while in the vector-valued scenario, think as though you have more than one features (say, weight, height, age of the individuals) available with you based on which your decision boundary will be decided in the classical problem of segmentation.
While working on classification problems, you need to make certain assumptions about the distribution of the features. Thus, you end up having the probability of the random vector taking a vector-value given that this point belongs to a certain class C [based on the joint density of any random feature vector conditional on class].