This is more vague than a proper axiomatisation, but the following procedure is broadly applicable.

1. Start with a space $X$ and a map $f\colon X\to X$.
2. Coarse-grain your space to a certain scale, so that orbit segments that are very close together are not distinguishable.
3. Count how many mutually distinguishable orbit segments of length $n$ it takes to be "significant"; call this number $a_n$.
4. Find the growth rate $\lim_{n\to\infty} \frac 1n \log a_n$; this is the entropy at the particular coarse scale you chose.
5. Let the coarse scale become finer and finer and take a limit to get the entropy.

Depending on how you make that procedure precise, you get various notions. For example, if $X$ is a topological space, "certain scale" means "code by an open cover", and "significant" means "covers X", then you get topological entropy. On the other hand, if $X$ is a measure space, "certain scale" means "code by a partition", and "significant" means "covers a set of uniformly positive measure", then you get measure-theoretic entropy.

This isn't a full axiomatisation, partly because it's a little vague, and partly because I only am really familiar with the notion of entropy in two contexts: topological space and measure space. Nevertheless, there's a commonality to the procedure in both those cases.

1. Start with a space $X$ and a map $f\colon X\to X$.
2. Coarse-grain your space to a certain scale, so that orbit segments that are very close together are not distinguishable.
3. Count how many mutually distinguishable orbit segments of length $n$ it takes to be "significant"; call this number $a_n$.
4. Find the growth rate $\lim_{n\to\infty} \frac 1n \log a_n$; this is the entropy at the particular coarse scale you chose.
5. Let the coarse scale become finer and finer and take a limit to get the entropy.

Depending on how you make that procedure precise, you get various notions. For example, if $X$ is a topological space, "certain scale" means "code by an open cover", and "significant" means "covers X", then you get topological entropy. On the other hand, if $X$ is a measure space, "certain scale" means "code by a partition", and "significant" means "covers a set of uniformly positive measure", then you get measure-theoretic entropy.

I'd be interested in knowing if there are other notions of entropy for other sorts of spaces that have analogous definitions. Or for that matter, if there are other notions that don't have analogous definitions.