Many discrete and continuous univariate probability distributions possess natural, straightforward and unique multivariate extensions/generalizations: the (negative) Binomial, the Gaussian/normal, the Student t, the Beta, the Laplace distribution, etc.

But many univariate probability distributions do no possess such natural and unique extensions: the exponential and Gamma distributions admit quite natural and unique matrix-valued multivariate extensions (the Wishart distribution). But they do not admit a unique and well-defined vector-valued multivariate extension. Same for the Poisson distribution, etc.

For instance, we find at least 23 different multivariate extensions of the exponential distribution:

23 Multivariate exponential distributions and their applications in reliability

And we find also many multivariate extensions of the Poisson distribution:

A Review of Multivariate Distributions for Count Data Derived from the Poisson Distribution

There are many, many papers like that. For instance, we already find more than 5 different bivariate extensions of the exponential distribution, etc.

I’d like to understand why please? To make it concrete:

Why do we talk about THE multivariate Gaussian distribution and why do we talk about 23 multivariate exponential distributions? Conversely, why don't we talk about 23 multivariate Gaussian distributions and why don't we talk about THE multivariate exponential distribution?

What’s the difference between uniquely-multivariatizable distributions and non uniquely-multivariatizable distributions? Can we classify them? What’s the obstruction to unique multivariatization? Is it analytical, algebraic, ...? Feel free to complete both lists.