You can use linear discriminant analysis. Let's say that we have stochastic variables $X$ and $G$ where $X$ signifies the measurement, and $G$ the underlying distribution (thus, $G$ takes values in $\{1,2,3\}$ in the example). The coordinates of $X$ are independent and furthermore, $X$ and $G$ are independent. Basically, we're interested in the quantity

$$
P(G = k | X =x ),
$$

where $x$ is a realization of $X$ and $k\in \{1,2,3\}$. For clarity, I'll just continue the example. The generalization is straight-forward.

By Bayes' theorem, we get that

$$
P(G = k | X =x ) = \dfrac{f_X(x|G=k)P(G = k)}{f_X(x)}.
$$

In your example, we have no prior information on the distributions, so let's assume that they are equally likely before we observe $x$, i.e. $P(G = k) = 1/3$ for $k \in \{1,2,3\}$. We can rewrite the denominator using our assumptions and using that the distribution of $X$ is a mixture of distributions.

$$
f_X(x) = \sum_{i=1}^3 f_X ( x | G = i) P(G = i)
$$

Thus, we get that 

$$
P(G= k |X=x) = \dfrac{f_X ( x | G = k)}{\sum_{i=1}^3 f_X ( x | G = i)}.
$$

These will be numbers in $(0,1)$ and they sum to $1$, thus it is tempting to see them as probabilities. In a frequentist set-up, let's consider how to interpret these probabilities. There is a true distribution. Hence, this distribution has probability 1 of being the true distribution no matter the outcome. Of course, we do not know which one it is, though. This is analogous to the interpretation of confidence intervals in a parametric model. In a frequentist interpretation, there is a true parameter value. When we observe data and construct a 95% confidence interval, this interval either contains the true value or it doesn't. However, when we repeat the experiment infinitely many times, we would in 95% of the cases get an interval containing the true parameter.

I would say that, if one were to interpret the above probabilities in a frequentist manner, one would have to interpret them in terms of repetitions of the experiment, all giving $x$ as the outcome. A Bayesian interpretation is perhaps more straight-forward, as degrees of belief in the different states. In this case the probabilities gives us the posterior distribution.