In a universe where the continuum hypothesis ($CH$) fails we can ask about combinatorial cardinal characteristics of the continuum, but in a universe where $CH$ is true no such cardinals exist so this study becomes vacuous.

Does a similar phenomenon occur at the countable level in a universe without choice? Specifically, are there properties which are true for finite sets but false for $\omega$ which are still true for the cardinal of an amorphous set, like divisibility as suggested here by François G. Dorais?

In a universe without choice we have the existence of amorphous sets and we can ask about their 'amorphous cardinals' which are incomparable with $\omega$ (thank you Asaf for the correction) and may satisfy nice theorems, but in a universe with choice there are no infinite sets whose cardinality is incomparable with $\omega$ so this study becomes vacuous in similar fashion to the uncountable case.

A possible candidate for characteristics smaller than $\omega$ could come from theorems in finite group theory that become false for countable groups, since it is possible to have a group structure on an unbounded amorphous cardinal as constructed by Asaf Karagila here.

There is an article behind a paywall published in 2010 that appears to touch on these matters but I can't access it; if anyone is familiar with its contents and willing to give a brief exposition it would be greatly appreciated.