What is the statistical model for a multi-label problem? In a setting with a binary $y$ like dog/cat, a reasonable statistical model is to posit that the probability parameter $p$ of a $\text{Binomial}(1, 0)$ distribution is some function $f$ of features $X$. This leads to many common machine learning approaches like logistic regression and neural networks.
In a setting with multiple classes in $y$, such as dog/cat/horse, a reasonable statistical model is that the multiple probability values $\vec p$ of a $\text{Multinomial}(1, \vec p)$ distribution is some function of features $X$. Much like in the binary setting, this leads to many common approaches like multinomial logistic regression and various forms of deep learning (e.g., convolutional neural networks for MNIST handwritten digits).
In a multi-label setting, such as identifying if a photograph contains a dog, a cat, or both, what would be the statistical model?
(To a large extent, I think I mean the formal likelihood, regardless of the functional form of expressing the probability of class membership as depending on the features, but I want to leave it a bit vague to allow for an answer that I’m thinking about it wrong by thinking of statistical likelihood.)
EDIT
The comments have clarified that an Ising likelihood works. With that being the case, how do the prior probabilities of the classes come into the picture? For instance, in a logistic regression, if I have $99$ $0$s for every $1$, I expect low probabilities of $1$ unless the features are extremely informative. In a multi-label setting, it seems like the prior probability of each class would be the ratio of that class to all possible alternatives, which I would consider to be zero: out of every possible sight there is to see, the probability of seeing a dog ought to be tiny or even zero (and the fact that we’re on Earth (for now) and near dogs is what allows us to see dogs with frequency).
It seems like something like this fails for multi-label classification.

Let's generalise your example by supposing that we have $K$ types of animal/object.  As you correctly point out, if we are dealing with observations where these types are mutually exclusive then we would have $K$ possible outcomes and so we would use some kind of multinomial model with $K$ categories.  However, if these animals/objects are not mutually exclusive then we would now have $2^K$ possible outcomes so we could now use some kind of multinomial model with $2^K$ categories.  In principle this latter case is not really all that different from the former case, and the same types of models can be used.  In your example of a dog/cat/horse model, you have $K=3$ so there would be $2^K=8$ possible outcomes.
Note that this type of model does not impose any assumed structure on the combinations of animals/objects in the outcome variable; it is appropriate if $K$ is not too large.  If $K$ is large enough that $2^K$ is too large to deal with effectively then we would probably impose more structure on the model to specify some kind of parametric relationship between the animals/objects occurring in an outcome.  At that point the two forms of model would become different.
